If my salary is $6500 a month, and I intern for three months, how much can I expect to lose to taxes?
The amount of taxes you'll owe at the state level depends where you live. Since WA doesn't collect income tax, you will not pay WA any income tax.
However, if your state of permanent residence is not WA, and that state does collect income tax, you are probably on the hook to pay your state of residence income tax based on your earnings in any state, including WA. Even though the state where you worked doesn't collect income tax.
Unless your home state and WA have a reciprocity agreement, but it's unlikely a state would form such an agreement with a no-income tax state.
Example: if I call Colorado home, but I spent 3 months working in Washington and earned $10k, then Colorado will insist I pay them 4.5% of those earnings.
You will pay federal taxes regardless. The amount may depend on your income over the rest of the year.