There are no infinite sequences; in fact, I'll show that there is no loop even on an infinite board. Thanks to Umlin for starting the proof, and alphacapture for getting me thinking about monovariants.
Terminology: I will say two squares are "in line" with each other if they lie on a common rank, file, or diagonal. A "direct" check is a non-discovered check; i.e. a check given by the piece that is moved. A "stationary" piece is a piece that never moves for the entire loop.
Preliminaries (observed by other people): To get an infinite sequence, you need a loop. This rules out captures, pawn moves, and castling. As Umlin pointed out, any king move must discover a check, but a king can't discover check while remaining in line with the opposing king. So if a king moves once, neither king can ever move again, which makes a loop impossible. Thus the kings never move. Since there are no captures and no king moves, every move must block a check. In particular, this means every check must be blockable; e.g. no double checks or knight checks. (Checks discovered by moving a knight are okay.)
Step 1: At any given time, consider the number of pieces that are either checking or x-raying their enemy king. (That is: queens in line with the enemy king, rooks orthogonally in line with it, and bishops diagonally in line with it, regardless of how many other pieces are in between.) Since the kings themselves never move, this quantity will only change by at most one at a time; namely, the piece that is moved. I claim it never decreases. If it did, this would mean we're moving a piece that is x-raying the enemy king to a place where it isn't attacking or x-raying the king. But a piece that is already x-raying the king can't discover check. This means that we are giving neither discovered nor direct check, contrary to our goal.
Since the quantity in question never decreases, it can never increase either if we want to form a loop. This means that we never move a non-x-raying piece to a square where it does x-ray or check the king. This has multiple consequences for different kinds of pieces:
First, queens can't discover check, so every queen move gives a direct check. By the previous paragraph, every queen move must come from an x-raying square.
For the same reason, if a rook or bishop ever moves into a direct check or an x-ray, it must have come from an x-raying square. But rooks and bishops can't move from one checking/x-raying line to another in one move, and staying on the same line won't be a check. So rooks and bishops (along with knights, as we saw earlier) can never move to or from x-rays or direct checks. In other words, every rook or bishop that ever gives a check or x-ray must be a stationary piece. Every rook, bishop, or knight move must be a discovered check, going from one non-x-raying square to another.
As a consequence of the above, every move, regardless of the piece, must come from a square in line with the enemy king: queens move from x-raying squares to checking squares, and other pieces always give discovered check. Every move must also end on a square in line with the friendly king (in order to block check). So in order to form a loop, every move must both start and end on squares that are in line with both kings. This significantly restricts the squares that we can use.
Step 2: Using everything from step 1, we now consider different cases for where the kings could be, and look at the possible moves in each case.
Case 1: The kings are orthogonally in line with each other. Then up to rotation and stretching, the board will look like this:
The marked squares are the only ones in line with both kings, so every move must be from one of these squares to another. (There may of course be stationary pieces on other squares to provide discovered checks.) In fact, no piece can move to one of the starred squares: you can't block a check by stepping behind the enemy king, and you can't discover a check from behind your own king.
White queens can't deliver checks on squares 2, 3, 4, or 6, since there's no permissible square on which black could block such a check. So white queens can only move between the squares marked 1, 5, and "-" (there may be more than one of the latter). Since there are no legal moves between these squares, white can never move a queen. The same goes for black. In particular, every move must be a discovered check, not a direct one.
If there is ever a check given along the line between the kings (labeled "-"), the checking piece must be a stationary rook or queen. Once this check is discovered, it must be blocked. The blocking piece can then never move again, as it has no way to discover a check through the stationary rook or queen. This can't happen in a loop, so there are no checks along this line, and thus no pieces ever move to or from it. So all moves are from numbered squares to numbered squares.
Next, since rooks and bishops can't move to or from squares where they check or x-ray the enemy king, there are no possible bishop moves, and the only possible rook moves are between squares 3 and 4. There are also no knight moves. (If we shrank the board by a factor of 2, knights could go e.g. between squares 1 and 6, but that would also have the effect of putting the kings on adjacent squares.) So at most one player will be able to shuffle a rook back and forth, and the other player will have no permissible moves at all. Therefore case 1 is impossible.
Case 2: The kings are diagonally in line with each other. Then up to rotation and stretching, the board will look like this:
(I'm allowing an infinite board, so I'm not concerned about whether there's actually room to discover a check by moving a piece from square 1, for example.) This case is ruled out by the exact same argument as in case 1, except with the words "bishop" and "rook" switched. (Note that knights could move e.g. between 1 and 4 if we shrank the board by a factor of 2, but again this would mean the kings are on adjacent squares.)
Case 3: The kings are not in line with each other. Say the kings are separated by a squares horizontally and b squares vertically, where a > b > 0. I've illustrated the case a = 4, b = 2 on an 11-by-11 board below:
A word of warning: this picture will look different if you change a and b. The differences that affect sliding pieces are as follows.
Squares 5 and 8 only exist if a and b have the same parity. Additionally, if a = 2b (as in the diagram), then squares 4/9 are vertically in line, and squares 3/9 and 4/10 are diagonally in line. If a = 3b, then squares 4/8 and 5/9 are vertically in line, and 4/9 are diagonally in line. Otherwise, none of these pairs are in line.
Again, every move must start and end on numbered squares. But note that white queens can't go to squares 4, 5, 6, 9, 10, or 11, as checks on those squares couldn't be blocked. Accordingly, the only possible white queen moves are 1-3-2 and 7-8-12. But those aren't actually possible: the moves 2-3 and 7-8 are parallel to the black king and can't be checks, while the moves 1-3 and 8-12 are parallel to the white king and can't block checks. So white queens can't move, and of course the same is true of black queens. In particular, all checks are discovered checks with stationary attackers.
Next, white bishops can only move to squares 2, 3, 4, 6, 10, and 12, since they can't move into checks or x-rays. The only possible moves among these squares are 6-4 and (if a = 2b) 4-10. The former is impossible because it wouldn't block a check. So the only possible bishop moves are 4-10 for white and symmetrically 3-9 for black, and these only work if a = 2b.
Similarly, white rooks can only move to squares 1, 5, 7, 8, 9, and 11. The possible moves among these squares are 9-11 and (if a = 3b) 5-9. The former is impossible because it wouldn't block a check. So the only possible rook moves are 5-9 for white and symmetrically 4-8 for black, and these only work if a = 3b.
It's a real pain to figure out what knight moves are possible. I'll spare you the casework and just give the answer:
- lots of them, if (a, b) = (2, 1);
- 3-5-10-8-3, if (a, b) = (3, 1);
- 3-9 and 4-10, if (a, b) = (3, 1) or (3, 2);
- 4-9, if (a, b) = (3, 2), (4, 1), or (5, 2).
At this point, we've effectively enumerated all possible king configurations and all possible moves that can be made; we just have to show we can't make a loop out of these moves. The following lemma helps with the casework:
Lemma: Whenever a piece moves to square X, there must at some point be at least two non-stationary pieces on the ray starting at the white king and passing through square X. (The same is true for the black king.)
Proof: If a white piece moves to square X, it must in doing so block a check which is given by a stationary black piece along the specified ray. It must move again later in order to form a loop, but before it does so there must be another piece blocking this check, as claimed. If, on the other hand, a black piece moves to square X, there must again be a stationary black piece attacking on this ray (so that the moving piece can later move again with discovered check), and there must currently be another piece blocking the check (so that the white king was not already in check before black's move).
In all cases except for (a, b) = (2, 1), there are very few possible moves (enumerated above), and the lemma is enough to rule them out. If (a, b) = (2, 1), then the board is as below, where squares 5 and 8 don't exist:
In this case, the lemma implies that no piece can move to squares 1, 2, 6, 7, 11, or 12, as each of these is the only permissible square on some ray extending from either the white or black king. So the only permissible moves in this case are 3-10 (for a knight), 3-9 (for a black bishop), and 4-10 (for a white bishop). Using the lemma again, no piece can move to squares 3 or 10 (as squares 1 and 12 are no longer usable), so we have no moves left. This is a contradiction, so a loop is impossible.