Did whoever came up with the message imply that enable this feature will heat up the computer? If so, he certainly communicated it a humorous way. Also, what specifically does it mean by "removes the depth limit"?
Well, it will keep your processor calculating which produces maybe 50-100 Watts of heat. (or however much your processor consumes)
By default on lichess Stockfish only calculates a certain number of moves ahead and then stops. However there's that infinite option which tells Stockfish to keep calculating until it's actively stopped.
Wow, that's a sophisticated bot. AI has come a long way.