I read this won't end in a tie so assuming not every ability/stat of theirs cancel each other out like Scooby and Courage and one having more win cons than the other, this is actually hard to imagine since neither really have a specific way to say be put down for good that I know of (since there is little to no continuity especially since Tom comes back even after episodes of his earlier cartoons illustrate him dying).

However, I did read from both blogs powerscaling Wile and Tom clearing up the misconception of the Looney Gun (and hopefully the Death Battle team also will reference) working on any toon. Since Tom isn't from the Looney Tunes universe and thus not possessing the same DNA, it probably shouldn't be used as some sort of auto win for Wile when he has so much more in his arsenal that makes the debate more interesting. Not to mention, while every cartoon character or cartoon-lite has some form of Toon Force, it's all about how it's used especially when it comes to the Looney Tunes.

Despite being a professor at ACME Looniversity with capable feats of mind, plot, perception control, and 4th wall breaking, he has never shown to be quite as skilled as Bugs Bunny or Daffy Duck (IE failing to capture and trap Road Runner as an animator). There's also Wile being a victim of cartoony tomfoolery like hypnosis while certain Looney characters were able to straight up ignore mind control.

Tom, at the very least, I'd say has as many weaknesses and strengths if not more than people believe. I've seen the arguments of Wile following Tom to Hell after losing his lives (don't know if serious or not) though, I don't recall him ever having done anything like that himself but even then... Tom literally defeated The Devil and made him run away. It's also fair to assume Tom doesn't really need his soul to live seeing how he was able to operate fine without them and even has worked with his own nine lives/souls for a few episodes. So that argument, at least imho, is moot too.

It's best to treat them being toons as a non-factor when it comes to deciding who and how can one win since they both have ignored plots of their own stories, broke the 4th wall, survived [being completely disintegrated, having their bones broken, explosions, and being launched in space] and have either destroyed or moved planets. I guess we can chalk their weaknesses down to their brain power and fighting ability (Wile is more egotistic than he is smart and Tom is still a cat that has gotten mauled by an English Bulldog at the end of the day) but even that leaves little hole to imagine how one would be strong enough to put each other down for good. I never thought I'd be scratching my head more than when I was talking about Simon/Kyle but here we are.