Broadly speaking, mortarless stone bridges have similar maintenance needs to their mortared counterparts, the primary difference being that mortarless bridges need more of the same maintenance. This is because stones in mortarless bridges are much more easily dislodged.
Weeds and Trees
No stone bridges benefit from heavy vegetation growth between stones, and mortarless bridges are particularly vulnerable in this regard. While in a mortared bridge the joints are more or less sealed up by the mortar, a mortarless bridge has plenty of nooks and crannies for foliage to take root in. Trees, of course, are the worst problem, as their roots can do an impressive amount of damage by displacing stones. However, heavy foliage growth can hold in moisture, deteriorating the stones. A careful pressure-washing treatment can be very effective for removing heavy weed growth between stones.
Besides the obvious concern of scour, mortarless stone bridges often are vulnerable to floods in other ways. A mortarless stone bridge usually relies on small fragments of stones to keep things stable. This especially applies to the arch. These chips of rock between the large stones that compose the bridge have an important function. First, they keep the large stones from rocking, thereby keeping the structure sturdy. Even more important, they keep the friction between the large stones high. Friction is what holds the bridge together, and friction is especially important in the arch. A flood sweeping over the bridge can wash out these stone chips, loosening stones and even causing arch stones to slip. Preventing the bridge from being overwhelmed by floods can be a little difficult, as this usually means the bridge is too small for the stream it is spanning. However, some ideas can be found here.
Slipped Arch Stones
Slipped arch stones can occur in any stone bridge, but are much more likely to occur in a mortarless bridge. A single slipped stone is likely not a problem, though it merits monitoring. However, if several consecutive arch stones slip down, the bridge likely needs some help. Such a bridge is weakened, and may need reconstructed or removed from road service if the trouble is particularly severe. If the slipping of the arch stones is indeed caused by loss of friction between stones, it may be necessary to convert the bridge into a mortared structure by injecting grout into the joints of the bridge. This solution is sometimes the best way to stabilize a mortarless bridge.