"Don't come on too strong" means he/she is the one who is the pursuer. You would be telling them not to flirt/pick up on someone in too forward or too bold a way.
In this case, "come on" (to someone) means to attempt to interest someone romantically or sexually. "He was trying to come on to me, but I found him unappealing."
"Don't come off too strong," and "Don't come across too strong,"could work for the one who is pursuing or the one being pursued. Other ways to say it might be: "Try not too come off like a player." "Don't come across as too eager."
In this context, "come off (as)" means to appear to have a particular attitude, intention, or character. "I didn't want to come off as weak."
In this context, "come across" (as) means "to appear to have a particular attitude or character." "Sometimes he seemed like a good old pal, but other times he came across as an angry and unpleasant man."
the definitions are taken from www.thefreedictionary.com. I just did a google search for each "come on," "come off," and "come across."