My character was affected by the dominate person spell (and failed the will save) and was given some orders. Does he have any recollection or memory of the orders after the duration of the effect expired?
Yes, He Does
The creature, in fact, remembers the entire experience, and, actually, the creature can't just forget it. The spell dominate person [ench] (PH 224) reads, "You control the actions... through a telepathic link," and that victims "resist this control, and any subject forced to take actions against its nature" makes a new saving throw." Emphasis mine.
There's no action for forgetting. So while a dominator can totally command the creature controlled by the spell to forget something, doing so won't have any more effect than commanding the creature to fall in love with the dominator, experience pain spontaneously, catch a disease immediately, remember its own birth, see something ethereal, or poop out a squirrel. Such actions are beyond the scope of dominate spells unless the creature's abilities permit them.
A very careful dominator could make it difficult for the creature to understand the actions the creature undertook while under the influence of a domainate spell, such as commanding the creature to close its eyes or cover its ears, but those are actions which can be performed.
The game must assume a baseline of human understanding of the universe lest game designers be forced to rewrite the cosmos as a legal document. Memory is a function of baseline human understanding, and memory editing just isn't an assumed ability of creatures, so unless the creature possesses an ability to do that--and he's commanded to--, he ain't forgetting his stint as a magic puppet.
While actually under the influence of the dominate spell proper, the creature's memory remains as functional as his mental abilities, experiences (personal and perhaps game-mechanical), and abilities allow. Properly speaking, recall isn't governed by any particular ability score (although Intelligence appears the most likely candidate, arguments for Wisdom and even Charisma ("My sense of self is so vast that anything I experience matters deeply to me" or whatever) can be made), so determining the extent of typical memory is in the DM's hands. However, unless steps are taken to make the creature's behavior while influenced by a dominate spell fuzzy (e.g. the enchanter commands, "I order you to consume booze until your memories of my commanding you are a blur," and then lets the spell run out), a creature should be able to recount anything he's done as an enchanter's puppet as accurately as anything else he's done as could be expected of a typical being of his scores, experiences, and abilities.
This also helps explain why vampires are frequently depicted as throwing the best parties.
By RAW the target remembers, unless something else happens to make the target forget. But I would argue that the caster can safely order the subject to submit to modify memory spells or similar effects, as this isn't obviously self-destructive.
RAW, nothing about Dominate Person says that the subject forgets about their dominated activities by default.
Mind-affecting spells are often a place where effects are subtle and ill-defined however, if the dominator explicitly orders the victim to forget everything they've done, I'd probably allow it to persist past the duration. If for no other reason than you want a vampire's dominated victims to be pleasingly unclear on what happened...