During the proceedings, the fact that a dating spouse is already separated will be noted, but that does not necessarily mean the circumstances of the new relationship will not be considered.
For example, the judge might disapprove of the dating spouse's behavior and develop a bias against them.
Or, the other spouse may simply suffer anger and hurt as a result of the limited amount of time it apparently took the dating spouse to recover and move on.
Either way, the other spouse may become confrontational, may become unwilling to compromise and obstinate during the proceedings, or, at best, may become cold and distrustful of the dating spouse.
This depends on your state – most states no longer consider adultery a crime, but some do.
The other spouse, if they are not dating, may develop the idea that the dating spouse was committing adultery even if that idea hadn't surfaced before.
When one spouse is dating, the other spouse is likely to be resistant to shared custody agreements as well.
A divorce and the prospect of a new partner replacing them is often too much for a spouse to cope with, and may cause disagreements and unwillingness to come to a compromise with regard to custody arrangements.
In some states, the spurned spouse can sue for "alienation of affection." An alienation of affection suit is when a spouse who was cheated on sues a third party essentially for stealing the other spouse and breaking up the marriage.
The cheating spouse and the third party do not necessarily even have to have a sexual relationship – in some places, a family member who convinces one spouse to leave the other might be liable for alienation of affection (though this is very uncommon).
In many states, your date of separation has legal implications.