You do not need exotic infinite mass to move forward in time.
For forward time travel, we only need to stop our perception of forward travel while we are doing it, and stop, or slow the chemical processes of aging while it is happening.
This is essentially what happens with the Rip Van Winkle legend, and all speculations of suspended animation.
There is no physical law that prevents suspended animation from being achieved. It is essentially a difficult problem in chemical and biological engineering. We do it for short periods with cooling the body while performing some operations.
The hard part of time travel is traveling backward in time. That is a very difficult, maybe impossible, problem.
There has to be time travelers visiting us from the future. How else would all the history revisionists get their ideas?