IANAL, but I have read that lying to a federal agent is a felony even if you aren't under oath. Perhaps one of the legal beagles can weigh in...
I am not a lawyer. Lying about identification is definitely obstruction of justice. Other lying to police may or may not be obstruction of justice. You do have Fifth Amendment protection, but in general, it's a bad idea. It's different than perjury, obviously, but it can be pretty grey. On the other hand, cops absolutely can lie to you. To a point, if they want the results to be used in court. They can lie about anything up to the imaginary line of "coercion", which is essentially but not inclusive to death or immediate harm. There's also some minor formatting they can use to threaten death or harm, and it'd likely be usable in court. "I'll kill you" or "You will get the death penalty" is no-go. "You know, it's sad, but X percent of prisoners die in prison. Do you want to play those odds?" is a legal quasi death threat that could pass, depending on your lawyer.
Lying or telling the truth to local, state or federal LEO should never be an issue anyways because you should never speak to an officer about any significant or serious matter without a lawyer present.
One concrete and interesting bit. You must specifically invoke your right to remain silent and right to counsel. If you remain physically silent, it can be used against you. If you're not physically silent and verbally request your right to remain metaphorically silent, it can't be.
