Shimano thumb shifter
 

...on my Diamondback hybrid has become extremely difficult to shift to the large chainring. It's a Tourney model FD.
Along with being very hard to push, it also has a long range of motion. It's become almost impossible to shift while riding. It indexes easily back down. I tried adjusting the cable tensioner and small adjustments on the high and low screws without much success. I didn't want to try anything more involved without checking here first.

remove, clean, re-oil and put it back on yet? and see if you need to replace the cables and housing.

disconnect the cable at the shifter. while watching the fd, note the resistance when pulling & guiding the cable back & forth. A cable housing can cause your described issue.

Possible causes:
FD adjustment (limit screws)
bent derailleur cage, or mount
cable friction
failed cable or housing

Sounds like you're checking the limit screws already. Do I understand correctly that the FD can reach the big ring if the H limit screw set properly, but for some reason the force required to shift it there is very high?

Shifters can wear out (usually worn plastic internals), and Tourney shifters are less than $20 each.

Just trying to diagnose that the shifter is indeed the problem. I.e., a new shifter won't solve the problem if it's caused by improper set up/tuning.
Park Tool has a good guide on their website: https://www.parktool.com/blog/repair...eur-adjustment. Best of luck.

I'd start by seeing if I could isolate the problem to the shifter, the derailleur, or the cable run.

Does your bike have a bare cable run along the frame's down tube or top tube? Shift into the lowest chainring, then pull the cable by hand, away from the frame, like you're drawing a bow and arrow. Does the derailleur move? Does it move back freely when you release the cable? (If you can remove the chain, it may be helpful to do so in order to observe the derailleur over its whole range of motion.) If the derailleur moves easily enough when you pull the cable and returns on its own, then it's probably OK. The lower cable run is probably OK, too, which means we can move on and look at the other half of the equation.

To troubleshoot shifter operation, I'd want to know a little bit more about it. You mentioned it's a Tourney shifter, but that unfortunately doesn't narrow it down too much. There have been all sorts of Tourney-branded shifters. Regardless of what kind it is, you can try isolating the shifter and cable. One way to do that is to disconnect the cable from the front derailleur. Then, while pulling the free end of the cable with one hand, try to shift with the other. Between your two hands, you should be able to feel if it requires excess force to shift.

One thing you don't want to do is fiddle with the limit screws on the derailleur willy-nilly. Adjusting them wrong can cause your chain to derail completely, dumping the chain off the chainrings. A derailed chain can jam and cause other damage, particularly if it falls off toward the inside. If your front shifter is indexed, maladjustment can cause other shifting issues. For example, if you set the high limit too far inboard, you can make it excessively difficult or even impossible to shift into the big ring. (Come to think of it, kinda sounds like what you're describing, but you'd need to troubleshoot some to know for certain.)