Should I turn my boyfriend into the police if I know he stole from someone’s house?

That is your decision but if you benefit in any way, you could wind up in trouble, too. You could also get into trouble if the police find the property in your home.

You could be prosecuted as an accessory to the theft, but only if you actively help to conceal it. Simply doing nothing, including not informing on your boyfriend, would not make you an accessory. Making false statements about it to the authorities, for example, giving him a phony alibi, would probably be enough. In any case, it's never a good idea to talk to the police about something like this without a lawyer. You might get yourself jammed up.

If you accept or benefit from the items while knowing they were stolen then you get in trouble. Why would you want a boyfriend like that he will get you in trouble.

You do not have a legal obligation to report your findings.

You should get another boyfriend, this one is a burglar. If he will steal from neighbors he has no morals and he will treat you no better. If you stay with him you will be the next to suffer the consequences of his lack of good decision making or even violence. Those who sleep with beggars and thieves are no better. It was obvious that this was the only decision to make, how do you not see that? Do you want to date a man who breaks into other people's houses and steals? I will not even bother to advise you on whether you should turn him in, he might kill you or harm you if you did.

You could be charged as an accessory by not coming forward.