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I recently played a game on as black. My opponent and I transposed into, according to Lichess, the Russian Game, Damiano Variation, when white played Qe2 on move 4. I wasn’t sure how to reply, and I ended up losing my queen and resigning a couple of moves later.

Here is the whole game.

[FEN ""]

1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nxe4 4. Qe2 Nf6 5. Nc6+ Be7 6. Nxd8 Kxd8 1-0

As black, how should one respond to 4. Qe2 by white in the Russian Game, Damiano Variation?

It started to go wrong for you from the move 4...f6, which just walks right into Qh5+ completely winning for white:

 [title "why 4...f6 is a mistake"]
 [fen "rnbqkb1r/pppp2pp/5p2/4N3/4n3/8/PPPPQPPP/RNB1KB1R w KQkq - 0 5"]

 5.Qh5+ g6 6.Nxg6 hxg6 7.Qxg6+ Ke7 8.Qxe4+ Kf7 9.Bc4+ d5 forced to sac the pawn so Qg4 can be prevented (9...Kg7 10.Qg4+ Kh6 11.d4+ Kh7 12.Bd3+ f5 13.Bxf5#) 10.Bxd5+ Kg7 11.Bxb7 Bxb7 12.Qxb7 Nd7 13.O-O and white is completely winning, both on position and material count. 

General advice: Particularly against 1.e4 systems, you almost always should try to avoid moving your f pawn early on, as white's queen and light square bishop are activated early on and therefore kingside light square weaknesses can be easily exploited. Typical to the Russian/Petroff move order, is rather d6 instead of f6 in order to kick the knight back. Though 4...d6 happens to also have been a poor choice in your game, as in that specific variation 4...Qe7 is really the only reasonable move that maintains the balance of the position. So let's look at that next:

 [title "4...Qe7: cover your king and set up a pin on e5 knight"]
 [fen "rnbqkb1r/pppp1ppp/8/4N3/4n3/8/PPPPQPPP/RNB1KB1R b KQkq - 1 4"]

 4...Qe7 5.Qxe4 d6 6.d4 dxe5 7.dxe5 Nc6 8.Bb5 Bd7 9.Nc3 O-O-O

And black has a much more reasonable play than they did in the aformentioned line.

Generally speaking, inherent to this opening's move order, moves such as Qe2 and Qe7 come quite naturally as the e file is opening up quickly and the discovery threats have to be parried, and these queen moves allow you to do so with tempo. Moreover, as you can see from the last line we looked at, trades are abound in this variation, and as black, you should not worry about dropping a pawn if you manage to compensate with development and piece activity (specially in short time control games). By sacrificing the d6 pawn on e5, we found a target pawn on e5 to coordinate against, which resolves your development plan (in contrast to white's play, hesitating as to whether defend the pawn or to regain initiative), and you have a slight initiative after the long castles.

In Damiano variant you respond 4. ... Qe7 to 4. Qe2. Yes, white takes the knight faster than you, but then you simply make d6 move, and if white knight retreats, white loses Queen. So to black's 5. ... d6 white have to respond with 6. d4, then: you take knight with the pawn and white makes a move dxe5. After those there's a bunch of possibilities that are better to be all checked on lichess.org via its analysis function.

Not to create any complications, you may not beat the e4 pawn after white knight grabs e5. Instead 3. ... d6 is a much better move, in my opinion, making white's knight retreat immediately, after which you can grab 4. ... Nxe4 without a thought after 4. Nf3, for example. And here to 5. Qe2 you simply respond 5. ... Qe7 and thus are able to retreat your knight next move without having a hidden check.

Do not play 3...Nxe4 in response to 3. Nxe5.

I feel this important aspect is missing in the other answers. You can see the primary reason it doesn't work in your game - you end up losing the queen. Given that you can't just move the knight after 4. Qe2, your next-best option is to play 4...Qe7 5. Qxe4 d6 (or ...f6), which still results in you losing a pawn. It's not a disaster because Black does have some compensation in the form of superior development, but Black's position is still undoubtedly inferior.

Avoiding this line is the main line of the Petroff, and is much more stable. The main line goes 1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5.d4 d5. Black has gained a tempo (as seen from his advanced knight), but the knight is somewhat exposed and can potentially be forced to retreat, in which case White is up two tempi. In practice, White can usually achieve this, but needs to play pawn breaks like c4, which balances out. The Petroff is thought to be drawish for a reason.