Some people make a qualitative distinction between divesting from coal businesses versus divesting from oil. I'll concede at the start that coal is the worst, so if you choose to only divest from one sector then it should be coal. That doesn't serve as a reason to oppose oil divestment.
The main reason I hear for distinguishing coal and oil is that we can live semi-normally without coal but not without oil. Depending on the time frames, though, we arguably can't live without coal and we can live without oil.
First, the one thing that changes tomorrow is that the political willpower suddenly exists worldwide to stop using coal, completely, and the question you get to decide is when to implement that decision to maximize human benefit. I think it's pretty clear that while immediately ramping down coal use is a good idea, ending all coal use literally tomorrow would not be a good idea. The energy grid would collapse in many countries and take months and years to reconstruct, causing a deep global recession.
Instead you'd want to use a time frame - I'd guess the best balance of reducing emissions with minimizing economic disruption would mean 5 to 15 years for eliminating coal use in developed countries, and add five years for developing countries.
Second hypothetical is the same as the first except the political willpower also suddenly exists worldwide to stop using oil, and you get to choose whether and when to get to zero. In this case an appropriate time frame does allow us to live without oil. We switch the power grid to near-zero emission sources and switch transportation systems to electricity, maybe with some use of biofuels (managing land use impacts) and hydrogen. Maybe very limited, continued use of oil balanced by carbon-negative practices. All you need is time, maybe 30-50 years.
So no, we can't live semi-normally without coal tomorrow, but if that doesn't keep you from supporting coal divestment then it shouldn't stop you from supporting oil divestment. We can get to a point of living without oil, so it's not wrong to use oil divestment to push us in that direction.
I haven't talked about natural gas divestment, which is a little trickier. While the time frame arguments apply to it, there's also the fact that natural gas competes with coal at the same time that it competes with renewables. I think it's a close argument and I'd welcome the natural gas industry reaching a deal with renewable energy, but in the absence of that deal then I'd rather pressure the natural gas industry as well through divestment.
Finally, some other folks may lump in tar sand oil with coal divestment. I agree that tar sands are the worst oil option so it's a start, but not the place to stop.
UPDATE: thought I'd add that Palo Alto City Council unanimously passed a resolution calling on the state agency CalPERS, which manages Palo Alto pensions, to divest from fossil fuels. Palo Alto has done some pretty good work on limiting their own emissions, so that's evidence that divestment is compatible with direct emission reductions.