Generating a differential signal
I would approach this as two problems, generating a sine wave, and making a balanced line driver. Other answers have covered the sine wave generator, and it's an easy thing to research, and I have nothing to add there. However, I'll say some things about the differential line driver.

As some others have said, the canonical way to do this is with a transformer. Transformers work great, but are big and expensive. In audio applications you will need an even more expensive transformer to avoid introducing unacceptable distortion. However, if you want to look exactly like a dynamic microphone, this is your best option, since a transformer simulates more of the properties of the windings on a dynamic microphone than any other method.

However, any balanced audio signal you get from any modern device that is powered probably won't have a transformer these days, because of cost. Powered (condenser) microphones may fall into this category; mixing boards and preamps almost certainly do. I highly recommend you read Design of High-Performance Balanced Audio Interfaces for a survey of common techniques and a detailed explanation of the relevant concerns. Also see Balanced Transmitter and Receiver II from the same site.

There's one part of that later article in particular I'll summarize here: What's important is that the impedance of both lines is the same, so that noise will result in the same voltage, so that it can be rejected as common-mode. Having an opposite signal on the negative side doesn't matter at all. In that article, there's a schematic, under the section Hey! That's cheating:



See the article for detailed discussion, but you can plainly see that pin 3, the negative side of the signal, is just a connection to ground through a resistor. As it turns out, if you disassemble a lot of professional audio equipment, this is precisely the type of line driver they use. It's because it has quite a few advantages:


Simple
Easy to balance
If pin 3 is connected to ground on an unbalanced input, nothing bad happens


The only critical part here is making sure R2 and R3 are exactly equal. Use 1% or better resistors, or balance them with a Wheatstone bridge for best common-mode rejection.