Some non-compounds spelled th and pronounced /th/ rather than /θ/ are: Thomas, thyme, Thailand, and, sometimes, Neanderthal.
Many natives might tell you that the h is unpronounced—even though when they say the words, they pronounce it clearly. In fact, written t is normally pronounced /th/ even though the h is not written, and most natives don't notice. So, time and thyme are both pronounced the same: /thaɪm/, with the /h/. You can sometimes bring the /h/ to natives' attention by having them hold their hand in front of their mouth as they speak a word containing t, like tell or time or Thomas. The reason they don't notice the /h/ is that in English, /t/ and /th/ are allophones. Native speakers usually don't perceive the /h/ because they hear it as part of the /t/. The /h/ is normally omitted only when /t/ ends a consonant cluster, as in stem. The writing includes no convention for indicating the difference, and indeed most natives are unaware of the difference. The difference usually becomes perceptible to natives only in compound words. For example, if you pronounce posthorn without the /h/, it will sound wrong.
The people who pronounce Neanderthal with /th/ rather than /θ/ are mostly anthropologists trying to reflect the original German—in effect, maintaining it as a German word used in English sentences. As native English speakers, though, when they attempt this, they can't help but aspirate the /t/. Most people, however, fully Anglicize the word and pronounce the th as /θ/.