My suggestion is this: think about WHAT YOUR OWN CHARACTER would do. If everyone were to act accordingly to what's written on the manuals, then everyone would be basically a replica of an old dusty tome.
Is your elf/dwarf open-minded by nature or not? Is it good aligned? If yes to both, then by all means, try to give half-orcs who try to behave a chance.
Is he closed-minded, far more cautious, or does he have a personal grudge against orcs and orc-bloods? Then by all means, have him despise their ilk.
Just remember to give room to roleplay, and base your character's ideas on each individual half orc's behavior.
My cleric of Corellon just loathes everything orcish or orc-related, I do not even remember the last time she actually showed some open-mindedness (or worse, friendship!) towards an orc or an orc blood despite her alignment being Chaotic Good, as much as I don't remember her patron ever showing displeasure for her apparent loathing of orcs and half-orcs. Still, she could potentially help a half-orc doing a good deed should the need arise.
On the other hand, half-orcs are 9 times out of 10 the result of a male orc forcing himself on an unwitting woman, so she tries to be a bit more tolerant and lenient on them (that is, doesn't show open hostility unless poked and could even help one out if he demonstrates her of being honorable).
Remember, racial bigotry and racism is an entirely canon element in Forgotten Realms setting. But "living and letting live" is a prerogative of a Good Character. Loathe half-orcs as much as you want, but avoid giving them trouble unless they are asking for it.
Digression:
In a sourcebook there is a good quote on elven view
It basically says orcs do not deserve the emotion of hate for they are merely pestilence. Genocide is not to be committed to not stoop to their or the drows' level. They don't mind keeping population in check though. Also that doesn't mean welcoming attitude.
Link me the source please. As far as I remember, it says that elves
generally reserve their hatred only for their dark cousins, whilst orcs (and only of the full blooded variety) are simply seen as
pests and rodents to be eradicated.
Speaking of which, elves
generally have no qualms about killing orcish cubs or drow children. It is socially acceptable and no one of the Seldarine does really care (Shevarash actually encourages elves to kill as many drows as possible, save for eilistreeans, but most of his worshipers seem unaware of this detail and just keep killing every drow they see).
This does not by any way mean that they will derive pleasure in it, nor that they will boast about it, nor that they will try to exterminate every single orc in Faerun .
Simply put, each one they kill is an elf-slayer who has just been robbed of the opportunity of reaching adulthood and killing other elves. Racial preservation first and foremost.