'There are you and me' versus 'There's you and me'

Short answer

There's just your voice and mine.

This is perfectly grammatical for two reasons. The first is that when there and is is contracted the contraction can be followed a plural noun phrase. This isn't considered very formal but it is definitely grammatical.

Secondly, when the noun phrase following the verb BE is a coordination (when there are two or more noun phrases joined by the words and or or), the verb commonly agrees with the nearest noun phrase in the coordination. In the Original Poster's example this is the noun phrase your voice which is singular. For this reason the example is completely grammatical.

Full answer

There is versus there's

The existential construction takes there as a subject. There has no meaning, and often the verb takes its agreement from the complement of the verb BE. So if the Noun Phrase after BE is plural, the verb will be in a plural form. If the Noun Phrase is singular it will usually be singular:

• There is an antelope over there.
• There are some antelopes over there.

Notice, however, that in the examples above, the subject and the verb BE are not contracted. In normal speech these will nearly always be contracted. We will use there's instead of there is. It is also quite common nowadays to see them contracted in writing, and you can find instances in prestigious newspapers like the Times, for example.

Now when the subject there and BE are contracted like this, the verb doesn't need to agree in any way with the following Noun Phrase. So, for example, with regard to:

There is people with standards and people who have fun.

... this sentence would be regarded as ungrammatical by most, if not all speakers. However if we contract there and BE, then it will be grammatical:

There's people with standards and people who have fun.

This brings this sentence into line with Lennon's:

Imagine there's no countries.

Or sayings such as:

There's many a slip twixt cup and the lip.

Or usages such as:

There's times when I've wanted to box his ears

"Proximity agreement" in coordinations

Here's a quote from The Cambridge Grammar of the English Language (Huddleston & Pullum, 2002):

A further complication arises in existentials when the verb is followed by an NP-coordination, as in There was/^?were a bottle of wine and several glasses on the table. Were tends to be unidiomatic with an NP-coordination when the coordinate that is adjacent to it is singular, even though the coordinate as a whole (a bottle of wine and several glasses) is plural. Plural agreement, however, occurs readily in lists: There are still Brown, Jones, Nathan and Smith to interview.

^{CaGEL p.242}

In other words it is usually preferable - and it's also perfectly grammatical - to use a singular form of BE when the following noun in the larger noun phrase is singular. But when we have lists we can also use a plural form.

Of course, if you try to get your grammar advice from bad style guides, you may find that the authors have tried to regularise out the grammar into what they think would be logical. However, this does not make them correct, it shows that they don't understand what grammatical means!

Conclusion

Any reputable grammar such as CaGEL or reputable usage guide such as Merriam-Webster's Concise Dictionary of English Usage will tell you that when the Complement of the verb BE is a coordination of noun phrases, the verb will often agree with the nearest following noun. Very often if a plural form is used instead the results will be very bad:

• There are only one dog and two cats. (ouch)

If your natural language ear tells you that singular is better in a particular sentence, it is!

Note: This excellent post by F.E. gives a lot of very good info on this subject