Is there a (minimum) size limit for a quantum computer? I know that current computers can only go so small due to the Heisenberg uncertainty principle, but would this affect quantum computers?
Limit is not (only) the Heisenberg Principle, but matter quantization.
Current (best) integrated circuits are at ~7nm which is about one order of magnitude above distance between atoms in metals.
This means "conductors" in ICs are about 20 atoms wide and about 5 thick.
We are not far from minimum dimensions.
Next frontier is to stack several planes on the same die to gain space in the third dimension (this is already used quite extensively in memory chips).
I don't believe Quantum Computers will allow breaking these barriers.
OTOH Quantum Computing promise is to provide more powerful basic operations, which means they could achieve the same level of functionality with less active elements.
You're using "quantum computer" as if it is something that will replace today's computers. It's much more accurate to suggest that "quantum processing" will replace today's CPUs and memory (and even that's not entirely accurate).
Have you noticed that over the last 20 years the computational power of computers has grown to breathtaking proportions, but the physical box containing the "computer" hasn't changed all that much? You will still need a power supply, cooling, mass storage, and a means of connecting to input and output devices. The needs of interfacing with the human and storing information have always taken up the vast majority of volume represented by a "computer." I expect that mass storage will be dramatically improved with quantum technology... but our habit is to make the dang hard drive bigger so the price point stays the same. (Besides, tomorrow's games will always need more storage space than today's games.)
But the human interface part... the com ports and interace electronics that connect keyboards, mice, pads, monitors, printers, etc., it shrinks, but the needs will always exist, and they don't benefit from quantum technology. (I can easily believe the keyboard will only disappear after we create direct mind-to-computer interfaces, but that's just my opinion).
So, when you ask how small a "quantum computer" can be, the answer is "basically as small as we can build any other computer" because the only things likely to change are the CPU and memory — and we'll always want those to be bigger, better, and faster.