Tanzania is on the WHO list of Yellow Fever Endemic countries. Zanzibar is "safe". Moshi or Arusha is iffy.
If you are from the United States or UK (yellow fever free zones) you can theoretically enter Tanzania with no layover or recent trip to another yellow fever endemic area, you can "legally" enter Tanzania without a yellow fever certificate.
If you are only going to arrive at JRO and go straight to your hotel, then straight to Kili, then from Kili straight to your hotel, then straight back to JRO, you "probably" won't need it, according to the International Travel Health Doctor at the University of Utah.
But if you're going to do a safari, or hang out at the hotel for more than a day either side of your climb, it's probably better to get it.
Potential side effects include "mild flu-like symptoms" which I can attest to includes aches and pains and intestinal complaints that actually can impact your training. I heartily recommend getting it early in your training, at least two months before you head to Tanzania.
Not like I did and get it 11 days prior.