This answer will focus on chess as it is played, rather than subdomains like composition or engine development.
Castling is a move that usually accomplishes at least one of two things:
- It gets the king to safety.
- It develops the rook.
Or both, in just one move. As castling stands now, it requires one move to reach a position that would otherwise take three moves. This accelerates the game considerably, where a player can get king safety and an attack quickly.
[As an aside, historically, the current version of castling is not the first. There have been several others during the 16th to 18th century, like swapping the king and rook directly, or "free castling" (put the king and rook somewhere on the squares between them, inclusive.) They all had a slightly different flavour from modern castling and are interesting in their own right, but the effect is the same, returning to the two points above.]
The effect of banning castling would be to remove this acceleration of the game. However, it does not automatically make the game slow or boring. I draw from my experience of shogi for this part regarding openings.
- In the opening, the choice between king safety and rapid attack would be even more pronounced than before. This would of course radically change opening theory (move sequences), but also opening strategy (what to prioritise).
- Players could consider spending several moves adopting structures that allow the king to walk from the centre into safety in the wings. This multi-move strategy is termed "castling" in shogi (unfortunate name), and is a large and varied part of opening strategy.
- There would be options to forgo this "castling" in favour of immediate attack, leaving the king in the centre. This is clearly double-edged.
- If executing "castling", players would have to be cautious - the king remains fairly vulnerable for a while, and players have to maintain flexibility to avoid being caught off guard during this.
Of course, shogi is its own completely different game (drops make it more akin to crazyhouse chess than regular chess), but the general ideas behind its opening strategy would seem applicable to no-castling regular chess, hence why I refer to it in this answer.
What is certain though is that this change would have the desired effect: opening theory (move sequences) would need to be completely revised as many lines hinge on castling quickly. Opening strategy would also change to an extent, where the balance between king safety and rapid development adds another dimension to the game.
On the other hand, endgame theory would not change (barring a tiny handful of composed studies). Middlegame skill and general "chess skill" would be also largely untouched, perhaps even promoted in importance as the game would for a while rely more heavily on piece coordination and chess sense than on opening lines.
This no-castling change seems interesting and has the advantages of (a) not changing endgame theory, unlike the stalemate = win suggestion often brought up, and (b) not changing pattern recognition, unlike chess960 which yields unfamiliar positions from the get-go.