Why were the Mongols unable to penetrate into Southeast Asia?

I think some chronology might be helpful here. The Mongols did attack kingdoms in Southeast Asia, though they were spiteful in nature and very costly to the empire (Yuan dynasty). The forays into Southeast Asia began under Khubilai Khan during his reign over the Yuan dynasty. Several of these states were tributary states where the Mongols did not have the administrative power to subjugate the people groups and absorb them into the empire. The issue here is not necessarily that the Mongols were not able to fight the people groups in Southeast Asia due to the climate, the issue is that the Mongols did not have the manpower to administer these territories, they were spending copious amounts of money on the campaigns, and they had other military endeavors which were more important (i.e. the failed attempt to conquer Japan). We cannot dismiss the notion that the terrain was very hard to traverse and was not easy for Mongol troops, even troops from the Yuan dynasty, to cross. It is helpful to remember that one factor is not the only reason for why something was not successful.

The court of Khubilai Khan sent envoys to several countries to begin trade negotiations. In 1279 he sent an envoy to the Kingdom of Annam, the northern portion of modern day Vietnam. In 1281 he sent an envoy to Ceylon which is modern day Sri Lanka. In 1285 and 1290 he sent envoys to Malabar. Khubilai Khan sought tribute and trade. At this time, he ruled the Yuan dynasty in China. He wanted to make tributary states out of the smaller kingdoms in Southeast Asia. Morris Rossabi writes that Khubilai meant to recruit experts in the field of medicine, craftsmen, and interpreters.

Even before Khubilai sent an envoy to Annam, the king of Annam sent Khubilai Khan an envoy to present tribute. In 1279, the same year that Khubilai Khan sent an envoy to Annam, the king of Annam sent an embassy to Khubilai's court. Rossabi contends that this was under the guise of tribute, but it was most likely a trade mission.

In 1273, three emissaries were sent to the Kingdom of Pagan in Burma. The Khan's emissaries issued a request to the King for tribute. King Narthihapate executed the three envoys "for daring to suggest that he humble himself to the Great Khan" (Rossabi 214). In addition, King Narthihapate attacked a border state called Kaungai, which was north of Pagan. Khubilai ordered that the son of his trusted Muslin retainer be sent to Pagan to deal with the murder of the three emissaries. The son was named Nasir al-Din. It seems that this campaign was not successful. Nasir al-Din was only able to establish Mongol authority over 110,200 households along the Burmese border (Rossabi 215).

Khubilai Khan ordered another campaign into Pagan in March or 1280. In addition, Khubilai had trouble with the King of Annam (Tran Thanh-Ton) and the King of Champa, which is the modern day equivalent of southern Vietnam, Jaya Indravarman VI. Khubilai Khan asked that both of the kings visit his court on tribute missions, but neither obliged. Eventually Khubilai demanded that both kings send one of his younger brothers to the court, we think to serve as hostages.

In 1281, he instructed Sodu, his highest official in Canton and a Maritime Trade Superintendent, to lead a punitive expedition against the defiant and recalcitrant monarch of Champa (Jaya Indravarman VI) With an army of 5,000 men, a hundred ships, and a crew of able-bodied sailors, Sodu set out for the coast of Champa, a relatively short distance. (Rossabi, 216).

Unfortunately for Sodu, Jaya indravarman had withdrawn his troops to the mountains and guerrilla warriors picked off the Mongols. Khubilai ordered reinforcements to travel by land, but they had to travel across Annam to reach Champa. Khubilai assumed that he would be able to send troops across Annam with no problem.

Tran Thanh-Ton, the king of Annam, did not cooperate with Khubilai's assumption. Before Khubilai's soldiers could get to Champa, they were at war with the Annam. The Mongols were at first unbeatable and continued to push back the Annamese troops. They reached the Hanoi region, but were rebuffed.

Guerrilla warfare, heat, and disease began to take their toll on the interlopers. The Mongol-led armies were increasingly demoralized by the elusive enemy and the mounting losses of life and supplies. They were experienced in pitched battles, not in small-scale, unexpected engagements. (Rossabi, 217).

Khubilai's army was finally victorious when the King of Annam realized that he would never be rid of the Mongols. He knew that submitting to their hegemony in the region would be the only way to stop the Mongols. Both the kingdom of Annam and the kingdom of Champa submitted to Khubilai Khan.

The matter of Pagan was still unsettled and Khubilai Khan was absolutely determined to make the Pagan understand the wrath of the Mongols. In 1287, Khublai sent his grandson Esen Temür to Pagan. Esen Temür was successful in capturing the capital city and holding it for a few months. King Narathihapate was forced to send tribute to the Mongols in order to regain the capital.

The Mongols had gained little from this campaign, however. Burma offered tribute, but the financial cost of these expeditions was burdensome and difficult to justify. (Rossabi 219).

I hope this helps or at least gives everyone more information on the topic.

Morris Rossabi continuously cites a work by Sung Lien et al., Yuan shih, a work translated by Paul Pelliot which has narrative account of So-tu's expedition to Champa, several other works by the Yuan dynasty, and numerous works by C. C. Berg, John F. Cady, W. P Groeneveldt, etc.

Sources: Morris Rossabi's Khubilai Khan: His Life and Times, 1988. Jack Weatherford, Genghis Khan and the Making of the Modern World, 2004.