Is it ok to have 3 violin parts in a piece of classical music?

It really depends upon what type of orchestra you're writing for. An elementary school orchestra would almost definitely have three violin sections, whereas a traditional orchestra has two. However, you'd be suprised how versatile we are; an orchestra can adapt to pretty much anything. :) You might get some crap about it, but it really wouldn't be any trouble for a conductor to create a third violin section. He would just have to change the seating up a tad, at the VERY most.

EDIT: To add to what the answer below me has listed, my orchestra once played Vaughn Williams's "Fantasia on a Theme by Thomas Tallis", in which there are two seperate orchestras, each with two violin sections and 3-way divisi. Basically, that's 12 violin parts. As for the cello section, there are 2-way divisi for each of the first 3 stands and 3-way for the rest of the stands, making 9 cello parts in orchestra 1 alone. There are all sorts of inventive ways to orchestrate a billion parts at once.

(I have a rather amusing story about that piece, actually. In the Orchestra 1 cello section, there were only two of us trying to cover all 9 parts. It was quite a feat, but we almost did it. :] Almost. It involved a lot of faking and an obscene amount of thumb position.)

A typical orchestra has first and second violins, but often within those sections they are split. Sometimes a section is split in two, so the inside player on the desk plays a lower part while the outside player generally plays the higher part. Perhaps if you had the main tune on first violin, then the harmonies in the second violins split? I've played many pieces when the first/second violins split into 2, 3 and 4, often all in the same piece! Sometimes this is done player by player, sometimes desk by desk (the 2 people who share a stand play the same) Some pieces, eg the beginning of Rossini's 'William Tell Overture' has each cello in the orchestra playing a different part! If you're writing for a typical orchestra, they would have the traditional "first violin, second violin, viola, cello, double bass" parts, but within each section they can be split. If, say, you want the second violins split, you use the same piece of music for everyone in the second violins, just use a split stave and do a divisi. Sorry if this post was too patronising / not enough information - it's hard to tell how much you know already!

Is there a viola part also?

If not, can you rewrite the third violin part for viola?

That would make it more customary.

However, there have been exceptions.

Benjamin Franklin wrote a quartet for three violins and cello.

The Pachelbel canon is written for three violins, cello, and harpsichord.

The second movement of Schostakovich's Fifth Symphony has both violin sections divided in three parts, and the three parts in the first violin section are the same as the three violin parts in the second violin section. But there is a viola part, too.

I don't remember exactly how the Faure Requiem is scored,

but I know it isn't scored the customary way.

There is only one violin part which is marked Tacet for several entire movements.

If I recall correctly, there is more than one viola part and more than one cello part.

It might be easier on the performers if you decide which violin part you would like to have the most players.

Then give that part to an entire section.

Then give the other two parts to the other section, write the two parts on two staves, and mark it "divisi."

Then the outside player at each stand will play the upper part and the inside player will play the lower part.

There are usually (perhaps always) three violin parts in a violin concerto: the solo violin, the first violin section, and the second violin section. If you don't have a solo violin part, it would probably be a good idea to write the lowest of the three parts for viola, as another contributor has already suggested.

Unusual but ok to do. There are many pieces that divide the violins into 3 or 4 sections, some as many as every player on their own part.

You can either mark divisi in the 1st or 2nd violin parts or have a separate instruction on how to divide the parts.

Yes, it's okay. The violin section can adapt. You see three violin parts in plenty of pops-style music.