My sister has recently been using LogMeIn to access her laptop at home via a work computer. The family router, a Belkin N+ model, has been cutting out recently in the last few days and we lost our internet connection completely a few days ago. Google chrome displayed the message "resolving host" when trying to connect to a webpage and when I reset the router it appeared to have no assigned ip address. the modem and the router both displayed full connectivity but there wasn't any access to the internet
Unplugging the router and setting up the unit with the install cd seemed to fix the problem.
Could this remote login service be responsible for causing issues with the router's ip address?

Extremely unlikely.