The catcher fatigue system is designed to ensure that you limit your starting catcher to about 85-90% of total playing time. It does so by monitoring usage within a moving ten-day window as the season goes along.
If you're in a stretch where your team has no days off, your catcher will almost certainly get tired if you start him ten games in a row. Giving a catcher an off day once every ten days is sometimes enough to keep him rested, and giving him two days off will definitely keep him at full strength unless he caught a couple of long extra-inning games in that stretch.
A catcher's workload is determined on a batters-caught basis. In a modern season, a team typically faces about 6250 batters over a 182-day schedule. That's about 344 batters per ten day period, and because we try to limit catchers to 85-90% of total playing time, you should try to keep your catchers from facing more than 300 batters in a ten day period. You can monitor catcher usage using the Team Status report and the Status page of the player profile window.
If a catcher is used more than this, he will be less productive as a hitter and fielder, with the penalty being greater the further the catcher is over the limit. The penalty is very severe if you let someone catch every inning of every game, so it's not something you'd want to try on a regular basis.
Using a catcher at another position (including designated hitter) is equivalent to giving him the day off, but you need to do this in advance. If you catch him too much and he gets tired as a result, you can't play him at another position the next game without penalizing him at the plate. But if you play him somewhere else once in a while, that will help keep him rested in the first place.