In the second question (without 'the'), you are asking me if I have ability in general to do this. Am I strong enough maybe? This is because without 'the', 'cups' is just in general or all of them. In the first question (with 'the'), you are asking me to remove a specific set of cups - those on the table right now. 'The' tells me, your listener, that you think I know *which* cups you are talking about (the cups that are currently on the table I'll assume). Of course, if you don't say 'the', and I can see a table with cups on it, I'll know what you mean in that situation. I'll just know that you've made a language error and puck up the cups. But there are situations where it won't be clear and will cause confusion.

Claire's answer is very clear. Here is another example: 1. Have you walked the dogs? 2. Have you walked dogs? #1 only makes sense if it is clear from context which dogs you are talking about. In a family with two dogs, this is the normal way to ask if they have recently been walked. #2 asks if the person has ever in his life walked a dog.

the cups именно чашки, а не ложки и не скатерть

It doesn't really change the meaning, but it is less clear and doesn't sound natural when the article is absent.

Neither sentence has an unambiguous meaning, and the existence or absence of the article doesn’t change the meaning in a consistent way for every possible meaning. For example, if you’re asking whether it’s permissible for people to take cups from the table, then both sound natural and have the same meaning. But if I’m asking you to take all of the cups from the table, I have to say the first. It’s better to learn by seeing natural and unambiguous examples of the language.