Variant chess effect Variant chess is a relaxing break from standard chess. But in my chess club we are usually discouraged from playing such chess. I have  not seen any visible problem in my games. Can anyone tell me why variant chess is supposed to be bad for your game?
Well it's not necessarily bad for your game, but it can easily lead to confusion because the evaluation and the gameplay in general in chess variants are very different compared to standard chess. As a result, because you are in a completely different mindset, routinely playing these variants may make your understanding or judgement over standard chess games a bit rusty. 

tldr: Obsessively playing a variant may make your standard chess feel a bit rusty, and also consume a lot of time that you would otherwise be spending to study standard chess (say endgames...), specially if you are a serious club player. On the flip-side, playing variants can potentially make for good training of your calculation skills and attacking techniques/instincts, to name a few. But all in all, a healthy dose of them can be a lot of fun in fact, specially among friends :)

To explain a tad on what I mean by different mindset or gameplay: 

Let's take crazyhouse as our first variant of choice: In this variant, any piece you capture from your opponent goes to your pocket (and vice versa), which means that during your turns you can either make a normal move according to the rules of chess, or insert a piece into play from your pocket. First consequence is the fact that there will be no endgame emerging from this variant of chess, because the pieces keep getting reshuffled around and re-enter the game. On the other hand, any gambit opening becomes unplayable, becomes you'd be giving free pawns that your opponent can place now anywhere they need! Finally, how you assess a given position would have to take into account all the variations branching out from different piece insertions. So naturally, when returning to standard chess, you'll be left with a strange feeling at first, but I wouldn't necessarily argue that playing crazyhouse will have a negative effect on your play. Because if anything, it does teach in a very direct way how to exploit weaknesses by immediately setting an attack using your pocket pieces (for instance you'll get a lot of chances to practice different checkmating techniques such as the smothered-mate). Simple example below:

 [title "Checkmate in 2: white to play (pocket: 1 pawn, 1 queen)"]
 [fen "r1bk3r/ppppbppp/1p1N4/2n5/1pQ1N3/8/PPP2PPP/R1B1KB1R w KQ - 0 1"]


Solution:


   1.Q@e8+ Rxe8 2.Nxf7#


Three-check chess:

This is more of a extreme example where even though usual checkmating is still valid, the game becomes all about avoiding checks no matter what, because one can also win by giving 3 checks to the opponent king. So you will be exposing and sacrificing all sorts of good material just to prevent a check (and conversely to give checks) that in standard chess wouldn't be a problem to face. So again, you see the mindset with which you will play and assess positions will be widely different from standard chess (see example below). All to say that, hopping back and forth between variants and standard chess, will at first leave you with confusing feelings when returning to standard chess and your evaluation will become a bit shaky, but after some playing you will quickly re-adjust again and it all comes back again of course ;)  

 [title "Three-check variant, typical mis-play example:"]
 [fen "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]

 1.e4 e5 2.Nf3 Nc6 3.Bc4 h6 {`h6` would be a somewhat acceptable move to play had it been standard chess, to prevent Ng4, but in three-check it's a disaster of a move, because multiple forced wins in 6 moves follow for white from here! Try to find it one.}


Solution:


   One possible solution: 4.Bxf7+ Ke7 (4...Kxf7 loses in 2, with e.g., Nxe5+ and Qf3+ last 2 checks) 5.Nh4 Nb4 {black's only attempt to delay the incoming two last checks by forcing a check with Nxc2+} 6.Ng6+ Kd6 7.d4 Nxc2+ 8.Qxc2 c6 {doesn't matter which move, the last queen check is unavoidable} 9.Qxc6+ {game over by 3 checks.}