I have some concern when playing King's Gambit as white but it's interesting. After:
[FEN ""] 1.e4 e5 2.f4 exf4 3.Nf3 Be7
What is the best move for white?
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The black threat of Bh4+ is not really ones, white should play Bc4 in this position. Releasing the f1 square for the king after the check on h4. After the check white cannot castle anymore but the bishop on h4 is not very happy, black can't play the g8 Knight and need to spend one more tempo on retreating the bishop.
If black don't give the check, white could just occupy the center with the d4 pawn, castle and capture back the f4 pawn with a nice game.
In this position, 4. Bc4 is played the most. It continues White's development, and 4. Bc4 makes room for the king on f1 (which isn't a bad square, at least not any more than h4 is a bad square for the black bishop). E.g. after 4. Nc3, Black can play the nasty 4... Bh4+, forcing the king to move to e2. The calm 4. Be2 works as well, but I feel it is less in the spirit of the King's Gambit, which is an aggressive opening.
4.Bc4 is a good move here. The line usually worried about, the bishop check 4...Bh4+, often leads to a line with strong, aggressive chances for White. Sacrificing pawns and using black's pawn for protection, 4...Bh4+ 5. g3 fxg3 6. 0-0 gxh2 7. Kh1 gives's White's king surprisingly strong safety as well as piece development and attacking chances.
As a regular 3, ... Be7 player, I can say that Be7 is not played to give an immediate check on h4. As others pointed out the best move is 4, Bc4 but white must be aware of the several tactics that could arise in this position. For example after the natural 4. Bc4 attacking the f7 pawn,
[FEN ""] 1. e4 e5 2. f4 exf4 3. Nf3 Be7 4. Bc4
Black would simply develop his knight to f6.
[FEN ""] 1. e4 e5 2. f4 exf4 3. Nf3 Be7 4. Bc4 Nf6
The idea is to attack the central e4 pawn and plan a trick if white tries to protect the pawn by 5, Nc3 then comes
[FEN ""] 1. e4 e5 2. f4 exf4 3. Nf3 Be7 4. Bc4 Nf6 5. Nc3 Nxe4
if 6, Nxe4 then 6, ... d5 wins the piece back with a fine position.
[FEN ""] 1. e4 e5 2. f4 exf4 3. Nf3 Be7 4. Bc4 Nf6 5. Nc3 Nxe4 6. Nxe4 d5
I have seen most players would give back the piece immediately after 5 ... Nxe4 by 6, Bxf7+ Kxf7 and then 7, Nxe4
[FEN ""] 1. e4 e5 2. f4 exf4 3. Nf3 Be7 4. Bc4 Nf6 5. Nc3 Nxe4 6. Bxf7+ Kxf7 7. Nxe4 d5
This is a perfectly playable position for black. The king could seek shelter in g8 after developing the rook to f8 or e8. The bishop at e7 guards the possible knight maneuvers and there is no real threat of decisive attack in this position.
There are many more interesting variations after 4, Bc4 but I am stopping here as the OP would likely be able to discover them without much effort.