NOTE: The Kvetching Editor wrote a response to this post here. I just want to point out that her situation is not what I'm describing. I'm talking about a single who enters the orthodox conversion process and then begins dating people who are already orthodox. However, for the record, I think dating is probably a bad idea during conversion for a majority of people, but there are always exceptions to the rule. My advice: Don't assume you're going to be that exception and seek out relationships. If they happen, they happen, but don't actively pursue it. That's my 2 cents. Now go buy yourself a stick of gum.
This is a touchy subject for many people. But this is the one rule you can expect to hear from every single conversion rabbi: No dating until after the mikvah (if you're currently single or become single during the conversion process). But why?
From the rabbis' perspective, it's generally an intermarriage issue. Especially today, most people who consider Jewish conversion don't finish. Therefore, even though you may be accepted to a conversion program, they hold out no particular hope that you will finish the process. However, if you're dating a nice Jewish person while converting, then drop out of the program for whatever reason, it's unlikely that the Jew will dump you over it. Then the rabbis worry you'll continue dating and end up marrying as a Jew and non-Jew. Of course, you say, "That'll never be me!" But you never know that for sure until you're done, and neither do the rabbis.
Further, from your perspective, you should discourage any orthodox Jew from dating you until you've gone to the mikvah and officially converted. He or she will get more flack from the community than you can imagine. Most notably, especially when it's a male Jew and a converting female, they will assume that the relationship is not "proper" (as in observing shomer negiah, etc), and they will question whether the Jew is "really" orthodox. They'll assume that the conversion candidate just doesn't know any better, but that the Jew knows exactly what he's doing and is trying to date non-Jews without looking like he's headed towards intermarriage. I've known people who survived this, but it's not pleasant. And it's one more outside pressure you don't need on a relationship.
Similarly, if your new partner isn't orthodox but you're in the orthodox conversion process, that is enough to derail your conversion for "not being serious." Many people approach the conversion process because they're dating a nonobservant Jew, but the nonobservant Jew normally studies to become observant as you study to convert. If you begin dating a nonobservant Jew after you've started your process, the rabbis don't expect that the other person will begin becoming observant but will actually draw your observance back.
So what happens if you happen to meet someone "nice" at shul, etc, and you want to pursue something? Stay quiet about it. Even if the other person says something and it becomes a mutually-acknowledged crush, most potential dates aren't willing to stick around for the time and pain that a conversion can involve (and most born-Jews have no idea what that will entail). Of course, you're sure that they will do it for you. And they probably think so too. And maybe they will. My advice: they can wait on you, but not in any official capacity. As I like to describe it: keep them on your "shidduch radar." If he or she is still available and interested when your conversion is complete, you can talk then. That doesn't mean you must avoid each other or can't be friends. However, for your sake and his/hers, keep any continuing friendship above reproach. Gossip in the Jewish community is bad enough without feeding it situations and words that can be misinterpreted.