I’m not aware of any long-term inter-species relationship mentioned in the ME universe (games or books), unless it involved an Asari or Shepard. There is certainly no mention of inter-species offspring, hybrids or half-bloods. Again, with the exception of Asari.
A (romantic) relationship is certainly not impossible. The comments/answers already mentioned the prime examples from the games. The Human female in ME3 obviously caring for a Salarian. Enough to buy him expensive armor to keep him save. Or the Turian in ME2, trying to convince his Quarian friend that she should date someone compatible to her – which means him. And of course Tali and Garrus if none of them are involved with Shepard.
These were not relationships however, but rather the possibility of one.
So it’s not uncommon in the ME universe to find someone of another species attractive. But it’s also not the norm. Most species seem to keep to themselves, with the exception of the Asari.
For them, relationships with other species are the norm while relationships with the same species are looked down upon.
Asari are special. Not only are they in a sense universally compatible, they are also apparently attractive to most other species. Which is a curiosity in itself.
But even if there was such a relationship, interbreeding would most likely be impossible. It’s not really clear if any of the species have compatible sexual organs.
Salarians apparently lay eggs, for example. How would another species fertilize these?
Even if they are compatible in such a way, it’s not said that they could successfully mix their DNA.
Both Turians and Quarians are dextro-amino based. While they have this in common, it basically only means they can eat the same food. But they might be biologically incompatible otherwise. I believe this is never clearly stated.
And Krogans would most assuredly have tried to get offspring like this, if it were possible.
We know that humans can’t interbreed with other species from our planet (at least not that I know of). Otherwise we would have hybrids running around everywhere. So it should be even harder with a species from another planet which might have a totally different genetic makeup.
Asari have a unique method of procreation which allows them to circumvent such natural barriers.
They extract the genes of the father (can be male or female) and mix them with their own. And they don't even need to have intercourse to do this.
Edit: According to the wiki they don't extract the genes but use the father as a map to rearange a copy of their own DNA.
This is why Asari always give birth to Asari and not to hybrids.
And this is why they are in a unique position to have such successful relationships with other species.