The clusters /pr/, /br/, /kr/, /gr/, /tr/, /dr/, /pl/, /bl/, /kl/, /gl/ are all pretty common. But why are /tl/ and /dl/ missing? Is there any linguistic or historical explanation?
The simplest explanation is that Proto-Indo-European did not have words with resolute initial *tl,dl,dhl, and there were no sound changes that created such clusters in most of our dialects (there are dialects where kl became tl, but they aren't well-known).
A related question is whether there is anything in the grammar of English that reflects this gap. It is also claimed (Clements & Keyser 1983) that English has a syllabification filter that prevents [tl,dl] in the onset – thus no words with initial "tl,dl". Actually, there are some, sort of, taken from other languages. In a small handful of cases, we are presented with "tl,dl", the best known example being Tlingit (the initial consonant is ɬ); other examples are Tlaxcala, Dlugos or Dlugosz, tlayuda. I believe that it is traditional Alaskan English to pronounce Tlingit as [klɪŋkɪt] (every traditional Alaskan that I know does so), but I believe that you have at least a 50% chance that other speakers of English would pronounce one of those tl-words with tl and not an epenthetic vowel or conversion to [kl]. I also suspect that this is because of Nahuatl influence, and does not apply equally to "dl". By way of contrast, English speakers cannot handle initial [ŋ] as in Nguyen, nor initial [nd, nt, nk, ng, mb, mp] where a vowel is inserted (Nkomo → [ɪnkomo, nəkomo]). The "can you say it?" test is at best weak support for a synchronic syllabification constraint (since some speakers may pronounce Tlaxcala with [kl]).
There is better evidence that there is an active constraint against [tl] onsets in English. In words like "atlas", /t/ is glottalized or even reduced to [ʔ], depending on dialect, following a rule that affects /t/ in syllable-final position. The syllabification [æt.ləs] is to be contrasted with [mæ.trəs] "mattress", likewise "gently" versus "gentry", with glottalization in the formwe and not the latter. The reason for the difference is that given an intervocalic consonant cluster in a stressed-unstressed frame, the longest possible sequence of consonants becomes an onset, so tr becomes an onset. Under the hypothesis that [tl] is not an allowed syllable onset, we predict that only [l] can go in the onset of the syllable, and [t] must be relegated to the coda of the preceding syllable.
I cited Blevins and Grawunder (2009) in a recent answer (to the question Are /tl/ and /dl/ rare onsets worldwide?), and I think that this source has some relevant things to say that I will summarize here.
Diachronic absence seems "accidental"
Specifically, in terms of history, Blevins and Grawunder say that Proto-Germanic lacked *tl and *dl onsets (p. 284). Proto-Indo-European had sequences of *t or *d + syllabic lateral (examples given: "*dl.h1ghós ‘long’, *tl.-né-h2- ‘lift.present’ "), but a vowel was inserted before syllabic resonants in Proto-Germanic. Blevins and Grawunder also mention the possibility of "rare" *tl onsets, "as in *tl@-ro- ‘high’(Orel 1998: 214)" (p. 289).
There doesn't seem to be any particular historical change that we can identify as being responsible for the absence of onset /tl/ and /dl/ in inherited English vocabulary; e.g. I don't know of a class of native English words that used to have tl- or dl- which became something else through a sound change.
In contrast, there is fairly clear evidence in Italic languages for various sound changes that got rid of tl or dl. This is only directly relevant to English in that a substantial portion of English vocabulary is borrowed from Latin or a descendant language; if Latin or French did have tl or dl onsets, they might have been introduced to English by such borrowings, but since they didn't, that couldn't have happened.
Word-initial /tl/ in borrowings
Word-initial /tl/ does exist in a few Ancient Greek words/names, and my impression is that it may be used on the rare occasions when people pronounce these in English; e.g. the initial "t" in the name Tlepolemus (Tlepolemos).
user6726's answer mentions other languages that have served as recent sources of tl-initial words in English.
Possible explanations for a synchronic rule
In terms of possible synchronic explanations, see page 287 (bold parts are summaries of information from that source, non-bold parts are my own commentary):
Blevins and Grawunder say that the Obligatory Contour Principle could play a role: this is where the idea of a dispreference of some kind for clusters of consonants with the same place of articulation.
Obviously not all such consonant clusters are prohibited in English, but I think user6726's comment isn't particularly convincing as an argument against the relevance of this concept. The clusters sn- and st- start with s, which is known to show special behavior in combining unusually freely with a following consonant in English. The clusters dr and tr are indeed homorganic, so they seem more relevant.
There is a separate idea that /l/ in English is less sonorous than /r/, which could help explain why /tr/ and /dr/ onsets are allowed but /tl/ and /dl/ are not (Blevins and Grawunder cite Moreton 2002: 56–57 for this point).
"Sonority" by itself doesn't seem to explain why we still see pl, bl, kl, gl, fl clusters, though.
There is perceptual similarity between /tl/ and /dl/ clusters and /kl/ and /gl/ clusters.
I'm not super convinced by that last point: it seems a bit circular to me, since the perceptual confusion seen in studies of e.g. French and English speakers is certainly influenced by the phonotactics.
Word-medial /tl/ and /dl/
Of course, /tl/ and /dl/ do exist word-medially in English. Whether or not they are onsets depends on how you divide a word into syllables; unfortunately, there are a lot of different approaches to English syllabification.
In native English vocabulary, word-medial /tl/ and /dl/ arises at morpheme boundaries; e.g. in suffixed words such as wit-less, wet-ly, ghost-like or heed-less, friend-ly, child-like, or in compounds such as land-lord, wet-land.
There are also words where these sequences have arisen from syncope of a vowel in a word-medial unstressed syllable. This syncope is optional in synchronically inflected forms of verbs that end in /əl/, such as battling, settling, huddling. The word butler is historically derived from syncope, but can only has /tl/ in modern pronunciation. The "t" in "butler" can be glottalized, which may imply the syllabification "but.ler".
Word-medial /tl/ in classical vocabulary
Morpheme-internal intervocalic /tl/ is encountered in some words in the learned stratum of vocabulary that come from Greek. The example in user6726 's answer, atlas, falls into this category. Because of issues relating to how stress interacts with syllabification, I think a more relevant set of words for a discussion of English onset constraints is Atlantis, Atlantic, Atlanta (all stressed on the second syllable): as far as I know, no native English speaker pronounces these words/names with a tl- onset in the second syllable.
This might show a constraint against tl- as an onset, although it might just be accidental. English syllabification does not always maximize onsets, even before stressed vowels. For example, English clearly has pl- as an onset, but there may be speakers who use /p.l/ in the word "haplology": the American Heritage Dictionary gives the syllabification hap.lol.o.gy, although Merriam-Webster gives the syllabification ha.plo.lo.gy. (The distinction between medial /.pl/ and /p.l/ would be realized phonetically in terms of greater aspiration, or greater devoicing of /l/, in /.pl/.)