As you may already know Japanese has a lot of peculiar nuances. In this case it's somewhat contextual.
If you are out of reach of a box of tissue and you are asking someone to pass you a tissue then there is an implication that you would like a tissue. Therefore the sentence "ティッシュを取ってください" is more akin to "Can you pick up a tissue [for me]?".
On the other hand if you are gesturing and holding a box of tissue in front of someone the same sentence could also literally mean "Please take a tissue" as you rightly guessed.
Regarding it's usage in other scenarios perhaps it makes more sense to look at the root sentence "ティッシュを取り出す". Here we can make note that "取る" signifies the action of pulling/picking a tissue out of the box.
Therefore you can use "取る" to refer to any action that involves plucking or picking up something. For example "引き出しからフォークを取ってください" (Please take a fork from the drawer).
However if an object is simply being passed to you it would be strange to use "取る", instead we would use the term "渡す" (watasu - To hand over). For example "鉛筆を渡してください" (Please pass [me] a pencil).
It is possible to completely omit the action and simply say "ティッシュをください" (Please give me a tissue). This would make more sense in the context of a restaurant where you may not physically see them retrieve the tissue.
As a rule of thumb "取る" is generally used when a container of some sort is involved, but confusingly it can also be used to pick things up from a table, shelf, etc. For new learners trying to navigate casual conversation it may be useful to omit these action descriptions that often have to be memorized.