Does an onboard GPU make the CPU hotter?

OK, there are a couple of things to check
1)  Are you SURE that the temperature readings are accurate?  If a processor gets "pretty hot" it has a tendency to shut itself down.  Are you experiencing heat-related shutdowns?  Before you answer this question, keep in mind that onboard (motherboard) temperature sensors are often wildly inaccurate.  If you haven't experienced any problems with your server other than suspicious temperature readings, I would first question if the temperature readings are accurate...

2)  For a server that is on all the time (and they usually are on all the time) they have a tendency to pull in a LOT of dust, and dust will cake up on your heatsinks and cooling fans and act as a thermal blanket.  Have you opened up the server to inspect it for dust?  If it's dusty, use compressed air to blow it out to clean it.  You can buy canned air if you don't have an air compressor.

As to answer your question more specifically, the onboard GPU (which means the chip itself is an APU, not a CPU or a GPU) will not cause your processor to run hotter, unless the chip itself is not cooled properly.  But then that points to a problem with the heatsink, which would most likely be DUST (see number 2).  Offloading the graphics (which aren't being used anyway) to a dedicated GPU would actually RAISE the temperature of the AMD processor.  That is because the dedicated graphics card would raise the ambient temperature inside of the case, as well as blocking / impeding airflow, which would make it just slightly harder to keep the AMD processor cool.

Technically it does but its MAYBE 1-2 C at max. 
a cheap passive gpu will heat your computer more (as they can produce much more heat)


just buy a better heatsink instead.

a better cpu cooler is what you want ... and cheaper, too

why would adding a GPU keep things cooler? wouldn't it be better to add a case fan?
and a passively cooled GPU doesn't mean it will be cooler. it in fact means it will produce heat, with NO BUILT-IN COOLER!
that doesn't mean it doesn't produce heat. it means it is cheaper, nothing more.
get a fan.

The integrated Radeon R5 GPU doesn't use much power as it is, so it's probably not the cause of your CPU woes. It's not like you have a GPU that pulls 100w or even 200w of which spits out a lot of waste heat. Most likely your heating issue comes from a problems with the CPU cooler, a weak CPU cooler, or the chassis has poor airflow.

Now, If I put a larger GPU in a case that has poor airflow then Yes! the CPU will run hotter. 

These old AMD A-Series APU's with 4-cores and a higher clock speed will run on the war side.

When you say
- it gets pretty hot
do you mean
A - it gets dangerously hot, hotter than it is supposed to get, hot enough to set off the system overheating warning alarm
or do you mean
B - yeah, I felt it, or yeah, I checked the temp with software, and it's hotter than I expected...but not hotter than it is supposed to be according to the specs, and not hot enough to set off the warning alarm

if A
then get a new, better cooling fan/heatsink combo

if B
who cares?

You say "It also has a Radeon R5 onboard GPU with it."
I assume you mean: a GPU combined with the CPU in a single chip.
(AMD A8 CPU, for example).

Likely adding a video board
combined with ***disabling*** the motherboard on-chip GPU in BIOS settings or by some other means
will indeed significantly reduce the CPU operating temp.

I can't say if you would see more improvement with a better fan/heatsink
or
if you would see more improvement with a new add-in video board and disabling the on-chip GPU.

Only by about half, keep your eye on it though.