Large trees often have roots protruding out of the ground around the base of the trunk (such as the ones that can push up sidewalks next to the tree). When is it safe to cut those roots (that might be a hazard to people running in the area near the tree), and when should you be worried about that causing damage to the tree?
There is some guidance here
Removing large or major roots from a tree is going to have a detrimental effect on the health of the tree i.e.
Any of those could easily lead to the tree dying.
Removing "sucker" roots, if done properly, should have no long term negative effects on the health of a tree, assuming they weren't sent up as a sign of stress (see below), even then they can be removed as the tree has much greater problems that need addressing (if possible, feasible to-do-so)...
Some trees by their very nature are prone to send up "sucker" roots eg
Other times "sucker" roots can be a sign a tree is in stress, and those roots have been sent up (above ground) as a survival mechanism eg
To remove "sucker" roots, use an appropriate sized pair of hand pruners or a pruning saw, make sure the pruning tool is shape and the cutting area is clean (use something like rubbing alcohol or a mixture of one part household bleach to 10 parts water). It's good practice to clean your pruning tool after working on each tree (plant, shrub, etc), as doing so helps prevent the spread of diseases if present. If you're really concerned about diseases being spread, you can of course clean your pruning tool after each cut.
Remove the "sucker" roots via a clean cut, either at:
Do keep in mind, just because you remove "sucker" roots once does not mean the tree will not send up more, in fact you might find the tree sends up even more...