I teach private lessons at a music store. Does the store have the right to tax the amount i pay them for rental space?

What kind of tax?  Is it a state sales tax?   What state?

or Property Tax?
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If you're referring to a sales tax, then yes, they absolutely do. In fact, is may not be just a right, but a legal requirement to charge and collect sales tax. In my state it is a legal requirement for a retailer to collect a sales tax on each individual sales transaction and pay that tax to the state. To top it of, the retailer is required to round up to the next penny. So with a 6% sales tax if you purchase something for $1, the retailer has to collect $1.06 from you.
Then the retailer has to pay 6% of that. Doing the math, 6% of $1.06 is $1.0636. Since the retailer is required to round up, they must pay the state $0.07 tax. Also, the tax is figured on each individual transaction, and not on the total of all sales in a period of time. So if there are two transactions for $1 each, and the retailer collects 6 cents sales tax on each, they end up paying 14 cents to the state since each transaction rounds up to 7 cents.
But regardless, you as the rent paying tenant don't separate that out on your books anyway. If the landlord is charging you $1000 a month rent with a 6% sales tax, that means you're paying $1060 a month rent, and that's the figure you will record in your books as rent paid each month.

Or, if you are referring to the store owner charging you a flat rent plus a percentage of your lessons, that is also legal. It all depends on whatever contract you have with the landlord. If you don't like the contract, you can cancel, renogtiate, or not renew it when it is up, depending on the terms.

@Opus 17
Why do you say it is illegal?   It is common in commercial leases.
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That should have said legal.  

Voice to text error that I didn't catch.

You are asking a legal question, rather than an income tax one.  Only a licensed attorney in your state can give legal advice..