The Republic of China, aka Taiwan, didn't fight in the Korean War because the United States didn't want Taiwan to fight in the war.

When the Korean War started, Chiang Kai-Shek, the leader of Taiwan, immediately offered to send 33,000 Taiwanese troops to Korea.

33,000 troops would have been the third largest troop size in the United Nations Command (UNC), the multinational military force that fought for South Korea during the Korean War. Out of all the countries that contributed to the UNC, only South Korea and the United States would end up sending more troops than this 33,000 number.

Initially, President Truman was inclined to accept this offer but changed his mind after Secretary of State Acheson warned that Taiwan going into Korea would provoke China to enter the Korean War against the United States.

There were also some concerns about the effectiveness of the Taiwanese troops, after losing the Chinese Civil War the year before. But, I would point to the presence of some of the other countries in the UNC who were also not as well trained as the American soldiers to argue that this decision was more about not drawing in China than about the effectiveness of the Taiwanese troops.

Before China's entrance into the Korean War, the United States was on the verge of winning the war and unifying the Korean Peninsula which shows that those 33,000 troops were not necessary insofar as China did not intervene.

At that moment, General McArthur, commander of the UNC, underestimated the possibility of Chinese intervention and the effectiveness of the Chinese army but even he said this:

Had they (China) interfered in the first or second months, it would have been decisive. We are no longer fearful of their intervention... Now that we have our bases in Korea, if the Chinese try to get down to Pyongyang, there would be the greatest slaughter.

Even a overconfident McArthur admitted that Chinese intervention, if it happened early in the war, would have been decisive. But, McArthur was wrong. China did enter the Korean War later on and when it did, it turned the tide of the war.

After China's entrance into the war, McArthur was in retreat and he wanted to accept Chiang's offer of 33,000 troops. According to McArthur, those Taiwanese soldiers were the only possible reinforcements that could arrive in the next two weeks.

But, Truman still declined the offer of 33,000 troops even after China has already entered the war and that was the original reason why not to employ those 33,000 troops.

Once China entered the Korean War, America was afraid that sending those Taiwanese troops would widen the war outside the Korean Peninsula. America feared that sending those 33,000 troops to Korea would weaken Taiwan's defense and open Taiwan up to a possible invasion by China.

Before the Korean War, China was getting ready to invade Taiwan although it still had to work out the logistical issues. China had amassed 156,000 Chinese troops across the Straits of Taiwan just a couple of weeks before the Korean War.

So before the Korean War, the Americans were getting ready to abandon Taiwan. Six months before the Korean War, Secretary of State Achelson gave a speech where he mapped out America's 'defensive perimeter' to include just Japan and the Philippines. And, Truman cut off any more military aid to Taiwan.

But with the Korean War, there was a growing appreciation for the strategic importance of Taiwan, and that it could not be lost too.

Sources:

The Joint Chief of Staffs and National Policy 1950-1951

The Generalissimo: Chiang Kai-Shek and the Struggle for Modern China