The only limitations on sheep hunting for non-residents are guide availability and being willing and able to pay for a guide. Those are barriers to entry, certainly, but there are plenty of folks for whom these things are not limiting factors.
Regarding who puts the most "pressure" on sheep, that is a tricky question to answer and it depends on how you define pressure.
If my memory serves correctly, in a typical year non-residents make up about 20% of the sheep hunters in AK, and they kill roughly half of the rams taken each year with an average success rate of about 65%.
Residents in turn account for 80% of the hunters in the field, kill about half of the rams each year, and have an average success rate of a little less than 20%.
Most guided hunters spend a week and change in the hills chasing sheep as part of their trip, whereas many residents can make a number of smaller, shorter trips throughout the season, hunt sheep in a few different places, etc.
Rather than pressure, I look at it more in terms of population impact, and it seems that res and non-res (guides) have about the same impact on the overall population each year.