Short answer: Nothing will break, you’re fine to open the doc on the new computer, and page numbers will continue to update automatically also on the new computer.
Explanation: LiveTOC uses InDesign’s own cross-reference feature. So when you run LiveTOC, it is simply turning the page numbers in your TOC into regular ID cross-references. Once that’s done, LiveTOC is no longer in the picture.
So you will be fine if you open the document on another computer and add or remove pages. The page numbers in the TOC will continue to update automatically regardless of whether LiveTOC is installed on that computer or not.
If, however, you need to add a new chapter or entry to the TOC, you will either have to turn that entry’s page number into a cross-reference manually, or switch to the computer that has LiveTOC installed and get LiveTOC to do it for you.