Tell me please if it is natural to say lie to me to a person when someone wants them to go up and lie with them in the bed, or would it be better to say lie next to me?
While the meaning you intend is "lie next to me", the context you give would make this sound very odd. It sounds like an oddly old-fashioned euphemism for "have sex with me". It is unusual for people to share a bed unless they are in a sexual relationship of some kind.
Whether or not you are asking about sex, such matters need to be handled carefully and clearly, so nobody gets the wrong idea. The expression "lie with me" or "lie next to me", while it carries the right meaning is not sensitive or clear.
There are (at least) two verbs "to lie":
lie(1): to speak falsely or utter untruth knowingly, as with intent to deceive
lie(2): to be in a horizontal, recumbent, or prostrate position, as on a bed or the ground; recline.
The phrase "lie to me" would be the first sense; in other words, "tell me something that is not true".
The phrase "lie next to me" (or "lie with me") is the sense you want.
(I won't touch on the wider problem of whether any of these are appropriate in any given social context!)
I'd like to add just a few words to the valuable comments given before mine. You might like to consider "lie down next to me" as well (of course, don't forget about the importance of the context you decide to use this phrase in):
Lie down next to me, look into my eyes... (from Take my Hand by Dido (0:15))