Ah yes, the effects of age, I could tell you a lot about that. But I’ll try to keep focused on in-game effects.
“I’ll regret this later”
Penalties to abilities that simply make the character weaker (like those found in older versions) are frustrating for a player, and they are bland, and don't really evoke the feel of old age. These rules simply never provided any fun, or much depth. It’s no wonder they didn't come forward to 5e.
Effects that limit the endurance a character has, that come into effect after some initial exertion, evoke the effects of age more poignantly, while letting the character still “relive past glories” in brief but glorious combat.
So very tired
The rules for exhaustion in the DMG can be utilized to simulate the fatigue of advanced age, and they come pre-play-balanced for you. A venerable character (depending on age, at the discretion of the DM) may gain a level of exhaustion from a single combat, and require a short or long rest to recover from it.
I just can’t seem to rest like I used to
Speaking of rest, an older character needs more. Simplest thing would be to double the amount of time needed to attain the benefits of the rest, but it doesn't add much color to the game, if the players are simply obliged to say “OK, then we rest for 16 hours.”
I find it’s better to “nerf” the effects of the rest, granting back fewer hit dice, hit points - or even fewer spell slots, if mental ability has been affected. That allows the oldster(s) to “try to keep up” while providing a nagging reminder they are really too old for this sort of thing.
(If you ever played 4E, this was like a non-heroic NPC taking a rest. Unlike the PC’s, the NPC would not wake up in the morning fully healed of all wounds.)
Another nice surprise
I would also suggest that these or any effects of old age should come as a surprise (especially for the prematurely aged). They always do.
My (game) experience with old age effects
I’ve used these rules for adjunct NPCs - I haven’t prematurely aged a PC. The players thought they were fair and interesting rules. It accentuated the power of the (young) PC’s, without making the NPC useless. The oldster slept/rested while the characters did ancillary stuff. It might be different for a PC. I suspect a player won't be overjoyed with having to “take it easy” but it's similar to other “curses.”
Since initially answering this, I spoke with one of my players about the age rules we used. She commented the rules made her feel protective of the NPC.
There are several means:
1. There is a priest spell in the tome of magic:
Priest 6th level: Age Creature (reversible)
The reverse of this spell, restore youth, permanently restores age that has been lost as a result of magic [...] reduces the age of the targeted creature by one year per level of the caster. [...] A subject cannot become younger than his actual age as a result of this spell.
2. In the Dragon Kings Dark Sun supplement:
And it specifically states the wish spell as an example...
Wizard 8th level: Protection from Time (abjuration)
When confronted with a magical aging effect (such as a wish spell [...]), the recipient gets a saving throw versus spell to avoid the effect [...]
This one is evil to the core (can even change one's alignment), but does the job.
Wizard 9th level: Vampiric Youthfulness (necromancy)
Once cast, the victim ages 10 years immediately. [...] The caster becomes one year younger.
Best Answer
There are no current, directly codified, effects of old age.
However. There are several spells and other effects 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 an example of a monster capable of aging a creature, we have the Ghost who can cause aging of 1d4 × 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 points out, we may find out the mechanical effects of aging in the DMG, but largely, these will be left to the DM's discretion.