TL;DR (too long, didn't read): The “to” is optional, and in this case, you have to reframe the sentence to remove it. Either way is grammatical by today's standards.
By today's standards, the “to” is optional. To some people, it sounds archaic, and to some people, it sounds unnecessary or off-putting. That is fine, and it is because this final “to” felt more at-home with older English, and there is a misconception that a sentence cannot end in a preposition. Let's look at your example again:
It is not feasible to dedicate attention to all the topics I would like to.
This sentence is short for:
It is not feasible to dedicate attention to all the topics I would like to dedicate attention to.
If you want to keep the same sentence structure, and still avoid the “to” at the end, you can do this:
It is not feasible to dedicate attention to all the topics to which I would like to dedicate attention.
This example just uses the word “which” to change the placement of the “to”, so it doesn't show up at the end of the sentence. Another thing that you can do, as mentioned at the top, is to omit the “to” altogether, and it is still considered gramatically correct. In this example, that can't happen, because the sentence structure suggests that you would like the topics, and not that you would like to dedicate attention to the topics.
It is not feasible to dedicate attention to all the topics I would like.
A solution is to reframe the sentence:
I would like to dedicate attention to more topics than is feasible.
This again dodges the problem as to where to put the “to”, and in this case, seems less redundant, without the “to which” phrase, which may sound very archaic in today's English.