Can I drive a Tesla 3 as an Uber?

yes, you can. There are more than you think out there.

Yes, Tesla 3 would qualify for an UberX, UberBlack, and potentially Uber Tesla if it is being offered in your city.  With the price starting at $35,000~$50,000 (source) and the cost of operation (i.e. electricity costs) claimed at $0.03~$0.06 a mile (source), it seems like it could be a great option.  Unfortunately, there hasn't been enough prior examples with Telsa 3s for us to know if it is a good choice for operating an Uber business.

For Tesla Model X, the model that has existed for a few years, there are prior examples.  Paul English, a co-founder of Kayak, is known to drive his in his free time around Boston streets.

Can driving a $80,000~$120,000 car as an Uber be lucrative? Perhaps if it's for UberBlack, potentially. 

No long shifts however: You also do need to keep in mind that Teslas can only travel about 220 miles on a single charge. This means that your shift can only be 4 to 8 hours, depending on how much driving actually occurs and how fast you are going.   Then, once you've exhausted your charge, it can take up to 8~12 hours to fully charge at 220V.  So you may not be working long hours!

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Uber Driver, CEO of TaxiFareFinder and Unleashed, LLC