Hot water out of cold water taps

We have hot water coming out of the cold water taps if anyone else is using cold water in the house. Flushing the toilet for example. If noone else is using water, it seems to be ok. I believe that it must be a single handle shower faucet, sink faucet or washing machine valve. If it is the shower, does that mean we have to tear out the ceramic tile or can we remove the hot cold handle/valve unit and then change the insides? All three showers have single handle controls and have been installed in the past year or two. They all have that anti-burn safety feature which doesn't seem to be working either. They all have fairly new ceramic tile. Please tell me we don't have to tear them apart!
Help!{Feel-bad-00020063}
Thanks for any replies