There are no current, directly codified, effects of old age.
However. There are several spells and other effecteffects that do cause you to age prematurely. The Monk would be immune to these. There are also basic guidelines on how the races age and when they become adults/typically die.
For andan example of a monster capable of aging a creature, we have the Ghost who can cause aging of 1d4*101d4 × 10 years with one of their features. While there are no direct effects, the DM can very well rule that this kind of effect would cause you to no longer be an effective adventurer.
As GMNoob pointpoints out, we may find out the mechanical effects of aging in the DMG, but largely, these will be left to the DM's discretion.