I'm planning on having several races in my novel, each of which primarily uses a different language. The idea is that all of these languages were at some point related, and all of them came from one ancient, dead language. The problem I'm facing is that almost all of these races still live very close to each other in terms of proximity, so I can't use geographic isolation as an excuse for each race to develop its own language. How can I explain each race having its own language without stating isolation as a reason (because all of the races are very close together)?
Several factors can lead to linguistic differentiation.
I know you said this isn't a consideration. But if you have a setting where transportation technology is primitive or expensive, most people will mainly travel to the other nearby villages. Going 50-70 miles from home would be rare for ordinary farmers, perhaps a once-in-a-lifetime event. Another factor that puts a damper on travel is constant warfare and instability in the region. People traveled more widely during the pax Romana than during the more chaotic period after the empire fell.
Difficulty travelling won't result in different languages at a distance of 50 miles, but it will result in a spectrum of dialects. Add enough distance and dialects cease being mutually intelligible and are considered different languages.
(All of the above assumes a lack of modern transportation and especially communication technology. Local dialects were reduced drastically with the advent of radio and television.)
For a long time, I thought there were five Romance languages: French, Spanish, Portuguese, Italian, and Romanian. Actually, from a linguistics standpoint, there are many others. For example, Galician is spoken in Spain but closely related to Portuguese (considered by some to be a dialect rather than a separate language). Occitan (southern France and neighboring parts of Spain and Italy) has aspects of French and Spanish as well as some features of its own. These "minor" languages are intermediate stages on the dialect continuum between the "main" Romance languages. But because they aren't the official languages of their countries, they have been stigmatized and suppressed. Occitan speakers in France had to learn French; Occitan speakers in Spain had to learn Spanish. Prune the missing links between Parisian French and Castilian Spanish, and it looks less like a dialect continuum and more like two separate languages.
This is the most important factor. user61244's answer is spot-on: linguistic differences are magnified as a way of expressing group identity. Even if elves and goblins live in close proximity, they're going to develop different ways of speaking because no one wants to be accused of "sounding like one of them." The more two groups develop their own pronunciations, slang terms, etc., the less intelligible they will be to one another. When they have to understand each other, they will use a more neutral dialect (possibly a form of the original language, like the use of Latin in Europe); but at home among their own kind, they will use their preferred way of talking.
What's the backstory?
One question you should ask is, Why were these races speaking the same language in the first place? If they are different enough to be considered separate "races," that indicates they either can't intermarry (if you're talking fantasy races) or don't intermarry much. If they have a common genetic origin, it's a long time ago. Genetic drift happens much slower than linguistic drift.
Let me posit one scenario that could lead to different races speaking the same language: conquest. One of your races conquers the others and imposes their own language across their empire. However, the conquered peoples have their own, possibly unrelated languages, and continue speaking these languages among themselves. How long the native languages survive depends on how aggressively the conquerors crack down on them. Eventually, everyone ends up speaking the empire's language, but colored by their native language as a substrate.
When the conquered peoples gain their independence, they have forgotten their original languages and go on speaking their local versions of the empire's language. With time (and without a standardizing political influence), divergence is driven by the mechanisms above and you have separate languages again.
The Roman Empire and Romance languages are the obvious example for these phenomena. For a less Eurocentric but still pretty similar one, read up on Arabic. It was spread across a wide area by conquest, and has since developed into a dialect continuum. Some of the dialects are mutually unintelligible and might be considered separate languages. Educated people write and sometimes speak in Modern Standard Arabic, which is an updated form of Classical Arabic (the parent language) used throughout Arabic-speaking areas. People from different regions who need to communicate apparently use a mixture of MSA and their local dialects.
European countries are very close to each other. Smashed up along side each other, in fact. No wiggle room between at all. Yet there is French, Spanish, Italian. Romanian. Romansk. More if you count ones like Catalan. All from Latin. And it didn't take long - maybe 500 years? 70 miles is a haul if your roads are scary, unmaintained paths through the forest. Nature is a pretty good natural barrier. You can posit a dark age.
1) As part of enforcing self-identity and the fact they belong to different groups: career criminals almost universally develop their own language, best known example to me is "fenya" -- made up language in Russian prisons.
2) The large area taken by each race -- sheer distance plus relatively slow means of travel even in case of lack of natural barriers helps to develop dialects and subsequently languages
3) the fact that different parts of one race's territory borders different cultures: this differential influence can speed up development of regional dialects and move the language farther from it's common ancestor
The ancient tongue was created to be a universal language, usable and understandable by all the races. But after whatever government or power enforced it fell apart, the races began to adapt the language to better fit their own needs. Differences in the tongue, throat, vocal cords, and ears could change lead to changes in the language. A snake-like race may use more 's' and 'z' sounds in their words, and drop off 't' and 'd' sounds as they are uncomfortable for their tongue and jaw. Elves might be able to hear higher pitches, dwarves subsonics, etc.
And that is assuming the races do not have other differences in what or how they communicate. A visually acute race might use fewer, simpler words, relying on body and eye language instead. A race that is especially magically sensitive might develop additional words for elements of spellcraft that don't mean anything to other races.
Besides all the answers already given, would you be willing to accept the imposition of changes to a local dialect? A crazy, linguistically minded despot lording over a small group of subjects could possibly pull it off. He could decide that the dialect his people speak is wrong, or imprecise, or beneath his dignity, and decree that new verb conjugations be adopted, better-sounding nouns be crafted, etc. Maybe he wouldn't have what it takes to design a full-fledged artificial language, or maybe he'd like the language to return to its supposed roots (for example, imagine an English-speaking mad king who decided to purge English of Latinate words and others and leave only Germanic cognates in the lexicon). He wouldn't have to have a real knowledge of the old language; indeed, like most reactionaries and fundamentalists, he would probably guess wrong based on legend or on incomplete sources. That doesn't matter, really. For this to work it would take not only a very determined ruler but probably also a long-lived one and the help of a caste of linguistic purity enforcers (priests?).
Isolation isn't just a matter of physical distance. Two groups can live in the same town and still have very separate communities.
As unfortunate as it is racism may be the answer.
For instance segregation in the American South led to distinct accents, jargon, and slang. It's not hard to imagine that if the system had continued* on a long enough timeline that a distinctly different language could emerge.
* I'm aware that many communities are still heavily segregated and I hope that someday this can be remedied.
Do the races live in the same country? 'cause if not you just have to look at, once again, Europe. French, Italian, Spanish and Portuguese all come from the same dead language - Latin - but are different enough that, even though I'm Portuguese I struggle to understand or speak the other three languages. And while France and Italy have natural barriers, Spain and Portugal are neighbouring countries, otherwise separated from the rest of Europe by natural barriers - oceans and mountains. Nevertheless, Portuguese and Spanish developed to be very different languages and Spain even has different dialects in its own country, different enough that some argue for their separation. You could even have the example of China, where lower and upper classes speak different versions of what us in the West call 'Chinese'.
Basically, use politics.