Is it illegal for someone to lie about being a cop?

Yes, very.

Depends on what you do next.

CONTEXT is CRITICAL. If you lie in a bar as part of a pickup line, it isn't strictly illegal. If you act in a manner that would lead a reasonable person to believe you were an officer, and attempt to exercise the authority of an officer, that is a SERIOUS offense.

Yes, it is illegal to identify yourself as a copy if you are not. If you are a cop, however, contrary to popular belief a cop CAN legally say he/she is NOT a cop. So asking someone if they are an undercover cop is pointless.

Just saying you are a cop is not illegal.

It isn't a legal matter unless you order someone to do something or somehow take away their liberty by pretending to be a cop.

Mr. Goodhi has been claiming to be a cop on this site for years.

IMPERSONATION is FELONY dimwit!!!!

Probably. If they try to order you to do something then it's definitely a crime. If some guy in a bar says he's a cop when you ask him what he does he probably won't get in trouble.

Yes. It is a crime called : Impersonating an Officer.

Generally yes. In my state (Louisiana), it is the crime of "False Personation" if the person falsely claiming to be a law enforcement officer (LEO) does it in connection with any act one must be a LEO to perform or to obtain any benefit or privilege accorded to legitimate LEOs.

Yes of course.