There's some verrrry circumstantial evidence that COULD be construed as instances of Mario and Peach having some sort of love life.
The original Paper Mario starts off with Mario being invited to a party that Peach is throwing. That in and of itself means nothing, as many guests are attending the festivities and Luigi is also invited. But then Peach requests to see Mario in private, and that is kind of noteworthy, because secluding yourself from your other friends to spend time with your significant other is something that a couple might do. It is also interesting that the game ends with both of them, alone and seemingly deliberately away from everyone else who attended the parade, watching the fireworks. No retainers, no Luigi, just the two of them together in a setting that, if the two were dating, would be very romantic.
Then there's the opening to Super Mario 64, where Peach invites Mario to come visit her at her place and eat cake together. We're never told who else is going to come. Could be just the two of them.
And lastly, it is also interesting how, in Super Mario Sunshine, Mario is allowed to accompany Peach on her vacation. A vacation to which she's only brought a handful of retainers with her, and none of her other friends (like Luigi).
All of these scenarios have potential to be romantic in nature, and Peach could have been planning these situations as dates, before Bowser interrupted them. And since we only usually get to see the parts of these two's lives whenever something goes wrong (as otherwise it would make for a very boring game), it stands to reason that more of these types of get-togethers happen off-screen. And thus, provided they are romantically interested in each other, they would be dating.
Now, as I said, all of this is highly incidental. Every situation that could be read as a date could also be read as them just being very close platonic friends. There's no definitive proof for or against it, and trying to force a conclusive answer would lead us into the realm of complete fanwank.
Personally, I'm not too fond of this couple though. I think they're kind of bland together.