Part of the historical appeal of stone arch bridges was the fact that they needed but little maintenance. After all, stone does not rot out or need painted. Stone bridges also are substantially heavier than iron or wooden bridges, which meant that, whereas in major floods wooden bridges would set sail downstream and iron bridges slid off their piers, stone bridges tended to stay where you put them. That said, stone bridges still could use some routine maintenance to ensure the longevity of the structure. How much maintenance is required frequently is in proportion to how well the bridge was built.
Dealing with Pier Debris
A notable concern is the accumulation of debris against the piers of the bridge. While in the case of a stone bridge it is highly improbable that enough debris will accumulate to topple the bridge (though this technically is possible, if but rare), this debris will lead to constriction of the stream and can cause a serious scour situation. Small amounts of debris are not to be worried about (though debris tends to catch more debris) but when the debris begins piling up, it needs to be removed.
Dealing with Trees, Grass, and Other Green Matter
Another form of routine maintenance is removing growth from the bridge. Ivy is picturesque and is certainly by no means the worst offender, but ivy nevertheless holds in moisture, which, when combined with freeze/thaw cycles, is hard on stones. Ivy also tends to root in the mortar joints leading to mortar deterioration. Grass and other green matter can also lead to mortar deterioration, but the worst offender is trees. Trees (even young ones) not only completely wreck mortar joints, but as the roots grow larger also slowly lift the masonry courses out of place, causing significant damage.
The simplest way to deal with a tree is to cut off the top and apply an herbicide to the stump. Uprooting the tree is a very bad idea, as this can destroy massive sections of masonry. Applying herbicide to the tree will kill the tree, preventing further damage. Over time the tree will rot, allowing the masonry to gradually settle back into equilibrium. A more complex way of dealing with the tree, which may be the only solution in extreme cases, is to remove stones to get to the roots. The roots will need to be removed and the masonry rebuilt.
Dealing with Old Mortar Joints
Mortar joints decay over time. Repointing the joints will be necessary.
Repointing is pretty basic (even if tedious). It involves removing the decayed mortar and then adding new mortar into the joints. The main trick is not to use hard mortars, as these hard mortars can damage the stonework by fracturing the stones during freeze/thaw cycles.