I use a travel mug with a press built in and it takes me an hour or so to drink it. If I steep it the max 5 min. and leave it in the travel mug press, will the steeped tea become bitter?
French presses do not stop the steeping process. The press is simply a seive to filter out coffee or tea grounds. Any tea that stays in the press continues to steep. Pushing the press down all the way also will squeeze more bitterness out. So for most teas, I do not recommend leaving the tea in the press unless it's a Rooibos tea which will not get bitter. You might get away with a white tea that is brewed at a cooler temps and isn't as tannic.
For portable presses using most tea, I recommend taking out the press and using a standard mesh basket or disposable tea bag to steep, then removing the bag when finished. Bodum makes a press with a spare lid for this purpose. I also do not recommend interchanging coffee/tea since coffee leaves heavy residue, unless you scrub the press after using coffee with soap and water.
In general presses are ideal to make tea in - they provided maximum area for the leaf to infuse. In most cases, the tea should be poured out once steeped. There are a few Tea Press pots that are designed specifically for tea, and minimize oversteeping. I personally use a french press to brew my evening Rooibos and Honeybush teas .