In Jamaican English, why is there no θ sound?

For example, three is pronounced as "tree," and thing is pronounced as "ting." How come Jamaicans do not have a θ sound in their accents?

As mentioned in the comments, every language has its own inventory of sounds. English has a relatively high number of different sounds, and many of its sounds are rare in the languages of the world (the English "r" sound, for example, is very uncommon). When a word is borrowed into a new language (or a new dialect), it sometimes gets absorbed into the closest sounds in that target dialect. At some point, that just becomes the "correct" way to say something. In this case, it sounds like the voiceless interdental fricative just got moved to [t] (which is much more common across languages).

There are some versions of Jamaican English that contain the 'th' fricative. However, the phonology of Jamaican English is heavily influenced by the phonology of Jamaican Creole. Jamaican Creole has its origin with the people taken from Africa to be slaves in sugar and coffee plantations. Since African languages did not contain the 'th' fricative neither does Jamaican Creole and neither do some varieties of Standard Jamaican English.