For example, if the shares would have been worth $1,000 when you joined the company, then the strike price for the options is $1,000. If those shares would be worth $10,000 at the time you exercise, then you pay $1,000 to the company to receive the stock, and then you pay taxes on $9,000 (the difference) to the government. So you might be looking at like 3K in taxes in that case.
You owe the taxes for the current tax year. You owe the taxes regardless of whether or not the company succeeds. Company could fold right after you exercise and you'd still owe Uncle Sam 3 grand.
(Then there are also capital gains taxes to worry about after this. I am only describing your immediate tax burden.)
The only way to not be taxed at the time of exercise is if the share price hasn't changed since you joined the company.