Answer
Fine metal mesh can snag tarantula claws and is not the best lid choice for heavy terrestrial species that regularly climb the roof. A secure perforated solid lid is easier to make predictable.
Why snagging matters
A tarantula hanging from mesh can catch a claw or tarsal hook and struggle to release itself. A fall from the lid adds a second risk in a tall terrestrial enclosure.
Ventilation without mesh
Rows of drilled or moulded ventilation holes can provide airflow while giving the spider a smoother roof. Every hole must be too small for the spider to pass through.
Arboreal setups
Arboreals still need strong ventilation, but use a closure that is secure and does not create dangerous snag points around the retreat.
If you already have mesh
Check how often the spider climbs it, reduce unnecessary fall height and consider replacing the lid if snagging is possible.