Sounds like you strain them too much. This game is a lot more physically straining for tapping and aim than fps games. Make sure your wrists are pretty much straight, don't tense your hands. You might need to change grip/sens and tapping hand position to make it more comfortable and manageable.
Your hands will slowly adapt and you can abuse them more, but it requires a lot of straining and rest.
If you could play without pain after 5-10 min before, you might need a break to recover. There's many kinds of strain and pain, some is normal and some is alarming. Muscle pain/tiredness is the most normal and unalarming. Wrist/finger/joint/tendon pain or numbness isn't something you should ignore.