How long does a couple have to be seperated before they can have a relationship with another without it being used against them in divorce?

There is not a time limit for using it against the other spouse-- but in some states how that relationship functions can be used against the offending spouse.  Specifically, if the innocent spouse has been struggling financially while the other spouse has been expending community resources on the new girlfriend/boyfriend -- then some states authorize the courts to take this fact into consideration when dividing community property.  Essentially, they order that the offending spouse reimburse the innocent spouse for the excessive use of the community estate.

If the couple has children-- a similar rule can kick in with regard to the imposition of child support.  If the offending spouse is supporting new boyfriend/girlfriend's child (biological or step-child) with a lavish lifestyle, then some states allow the courts to take that fact into consideration when assessing child support.  The principle is that one set of a parent's children should not endure a lesser lifestyle than the other children.  As such, the court can increase the amount of child support-- even if the amount exceeds statutory guidelines. 

Bottom line, is that both spouses are better off keeping relationships at a minimal until temporary or final orders are in place that protect both sides.

Under the laws of all state sin this country (even one's allowing divorce premised upon fault) what one spouse or the other does after the legal separation and filing of a dissoultion action has no bearing upon what is decided in the dissolution. What matters is what happened up until the time of the filing.