Mostly they don't, but in the US they do. A Republican politician has lately started a campaign to change this.

The gold medal winners get $25,000 in the US, with smaller prizes for silver and bronze winners. This is a lot less than some countries, but it is still regarded as taxable income. There is a rightwing pressure group called Americans for Tax Reform which is claiming that the top winners could end up paying as much as $9,000. However, this would represent 35% of the total and would only apply to athletes who are already in the higher tax bracket.

The problem is the US's unusual tax system, which requires citizens to pay tax on money earned abroad both in the foreign country and the US itself. In fact the international olympic committee does not require athletes to pay tax on their prize money, but under US law it's just foreign earnings and is treated the same way as any other income.