Just like with a typical a cosmetic tattoo procedure, a cosmetic tattoo removal procedure should not negatively impact your skin in any way when done properly, much less require any significant recovery. Some mild redness and swelling should be expected over the next few hours and should greatly subside after the first 24 hours (applying cold compresses will greatly help). Keep in mind, however, that the location of the removal and amount of surface area of ink to be removed are major factors in any downtime that may be needed after a tattoo removal session.
The removal of a thick, dark eyeliner for example, should expect significant swelling of the upper eyelids for at least a day or so while the removal of a thin, faded old eyebrow tattoo should not expect much, if any, swelling. Any redness should be mild and last only for a few hours.
Above: old faded eyebrow tattoo
Below: immediately after tattoo removal
Below: Healed skin with noticeably lighter remaining ink