Does the inner tube affect the maximum PSI? If I have road tires that have a recommended pressure of 110 PSI and maximum of 120 PSI can I safely pump my tires up to that level or does the inner tube effect the maximum? 

I currently have the tires pumped up to 90 PSI however the question What Pressure Should I Run My Road Bike Tyres At seems to indicate I should go for the full 110. 

It feels when I'm pumping that it would be very hard to pump above the 90 mark and I am wondering if the quality of the inner tube (or a inner tube max PSI) could be the reason. Should I just go for it and try to pump it up to 110 PSI or am I best off leaving my tires at 90 PSI?
The inner tube has no significant effect on max pressure.  The valve stem area is the only area sensitive to pressure, and in general the valve can withstand 200 PSI or better.  (And if you exceed whatever limit there is the result is "catastrophic" failure, not simply the inability to pump in more air.)

What it sounds like you're noticing is the geometry of the pump.  A piston pump has a compression ratio, just like a cylinder in an engine, and the max pressure that the pump can generate is that compression ratio multiplied times the 14 PSI of atmospheric pressure.  In general, pumps are designed either to move a lot of air at low compression ratio to fill low pressure tires quickly, or to move less air at a higher compression ratio to be able to fill high pressure tires at all (though more slowly).

Good quality pumps will be advertised with a stated maximum pressure, and you can fill tires reasonably fast to 80-90% of that max pressure, after which progress will slow to a crawl.  So when you shop for a pump you need to pay attention to that number.