Can replacing my handlebar tape help reduce finger numbness? Despite changing hand positions often and switching to more "cushiony" gel gloves, I'm still experiencing intermittent numbness when I ride (sometimes I get tingling and sometimes not).

Are certain types of handlebar tape likely to help? The tape on my bike is several years old. I see gel and cork tapes (and those that claim to be both gel and cork), neither of which I have tried.

I realize that I don't actually know what the real purpose of handlebar tape is (perhaps that's another question). It might be that it is only for helping you keep a grip on the bars when they get sweaty.
Hand numbness or tingling can happen for a variety of reasons. It could be a matter of fit, or simply tape or the handlebars themselves. 

Bike fit: 

How's your riding posture? If you're putting too much weight on your hands, it could be because you need to raise you saddle to allow you to put some of your weight on your legs. It's also possible you just need to tilt it a litte. For example, I know that I put a little weight on my butt, even though I try to put most of it on my legs. My hands are happiest when I tilt my saddle so I'm almost sliding forward, but not quite. The point here isn't that you should do this; everybody's bike and body are different. (Well, some people have the same bike as others.) But minute changes can have a huge effect. 

Saddle height can also be a matter of trial and error. This answer gives the skinny on saddle height, although experimentation is always helpful. This question talks about bar height relative to saddle height. 

Is your bike sized correctly? If the distance from the saddle to the bars is too great (or too small), that could cause other problems (namely, making it difficult to get a good fit in the first place). You can get a rough idea of this by checking that your standover height is correct, although this is a very rough test. 

Bars, hands, gloves, and tape:

Angelo's answer gives a good overview of bar tape, and cork-vs-gel tape. In addition, I've heard that gel tape tends to wear out quickly, as the gel slowly migrates away from where you rest your hands. (Gel saddles have similar problems.) 

Another option is double-wrapping the bars; some people swear by it. 

You can also get padding for your bars that you stick to the bars before wrapping. I tried it and while I personally didn't feel any difference, some people love the stuff. 

Are your bars an appropriate width? A good rule of thumb is that they should be as wide as your shoulders, although this can translate into some very wide bars, particularly with stocky men. (Ahem.) 

Lastly, don't forget that padded gloves can help. Sometimes, when your hands start to hurt a bit, it just means that it's time for some new gloves. 

In conclusion:

Hand pain is, unfortunately, a problem that many cyclists fight for years; others lick it immediately or never encounter it. Don't be afraid to experiment with the goal of making yourself as comfortable as you can on the bike.