The Apollo astronauts' footsteps can remain theoretically forever on the Moon as it has no atmosphere. Mars does have an atmosphere but a very thin one. Its pressure ranges from 72 Pa (0.0104 psi) on the top of Olympus Mons to 1240 Pa (0.18 psi) in the Hellas Basin. Since this is quite a difference, I'm curious on both scenarios: how long would astronauts' footprints remain on Mars in case astronauts land on Olympus Mons, and in case they land in the Hellas Basin? Due to the low atmospheric pressures, the winds on Mars aren't by far as strong as on Earth.
Several months to years, but not forever.
We (obviously) don't yet have astronaut footprints to examine, but we do have rover tracks to look at.
As we can clearly see, over the span of one Martian year the tracks have faded a lot, but are still crystal clear.
Similarly, here is a close-up of the Spirit Rover's tracks during the 2007 global dust storm. In just 22 sols of severe duststorm, the rover's tracks have been 90% erased. You can still make them out, but they are mostly filled in and smoothed over.
Note that this was during an exceptionally severe storm, the like of which only occur one a decade or so. Mars does tend to have a lesser global dust storm ever 3 or so martian year, but the usual fare is not as severe as the 2007, 2018 storms were.
So realistically, expect wheel- and footprints on Mars to last 10 to 50 times as long as on Earth, but they will fade away in calendar time, not geologic time as on the Moon.
As Sujay says, Mars is known for it's dust storms, therefore footprints wouldn't last long on the surface. The regional Martian dust storms tend to happen in the summers and the larger, planet wide, storms every few years.
However, while I cannot find anything categorically stating the dust storms don't reach the top of Olympus Mons. The vastly lower pressure (around 30Pa compared to around 1200Pa on Hellas Planitia) - https://planetary-science.org/mars-research/martian-atmosphere/ - makes me think that they likely don't; and even if they do, it would mean they carry vastly less material and therefore, footprints on the top of Olympus mons may be able to last an extremely long time.