Could be a matter of just getting used to it (same as with saddle) as you are doing something different with your hands.
If you are putting too much weight on your hands you may well have tired arms too. Kyuss and tape above is one solution. I have seen newer bikes with a flat top to the bars to give a broader contact area.
This however is not addressing the root of the problem. You can get a bit less weight on your hands by moving your saddle back slightly (or sitting back a bit) as you may be tipped forward too much.
This is assuming the bike is about the right size for you.
Change hand position. I have drops but rarely are on the lower part. I tend to move my hands around between four positions on the bars.
Lastly, if you are new then you will be a bit more of a dead weight on a bike. As you get fitter, you will be putting a lot more of you weight onto the pedals and less on the saddle or bars so it may just go away.
