Before getting to the specifics of Telugu, it would be useful to review what it might mean for an orthography to be 100% perfectly phonemic. Based on how the notion "phonemic" is used in language description and orthography development, it would be a system where for each phonemes there is a unique and rule-governed orthographic sequence, and for each such orthographic sequence there is a unique mapping to a phoneme. I say sequence since in a language that does not have the cluster "s+h", we could represent the phoneme ʃ as "sh". Furthermore, this 1-to-1 correlation between orthography and phonemes is not tempered with exceptions of the type "except in the word..." or "depending on the origin of the word".
Here are some examples of features that would disqualify the orthography. In Russian, unstressed /a,o/ are neutralized to a and /e,i/ become i: this is not reflected in the spelling. When the phoneme /o/ neutralizes with the phoneme /a/, in a phonemic orthography, the spelling would change. In this respect, Russian spelling is not phonemic. (I have nothing to say about the letter ы because the phonemic status of that vowel is controversial). Additionally, stress, which is not predictable (is phonemic), is entirely omitted from the spelling. If phonemic distinctions are omitted in the spelling (stress), the system is not purely phonemic. If orthographic distinctions are made which do not correspond precisely to a phonemic distinction (vowel letters are distinct where the vowel sounds are not), the system is not purely phonemic.
One other potential disqualifier would be if there is a distinction in letters which is in all contexts predictable – it reflects a phonetic fact which is not phonemic – then the system is not purely phonemic. A hypothetical example would be if English spelling notated aspiration with a raised ʰ – there is a phonetic difference, but not a phonemic difference, so it should not be indicated in the spelling. It is possible that Sanskrit has that property, since there are graphemes that reflect sub-phonemic differences (or even possibly non-existent sounds). The problem is pinning down a specific historical stage of Sanskrit pronunciation and writing: an examples is "ñ" which is predictable (adjacent to a palatal) and probably does not exist as an independent phoneme once you exclude later introductions (the noun ञ, plus grammarian's meta-usage). "Jh" (झ) is at least suspicious as a phoneme.
One very common feature of writing systems frequently renders the system not 100% phonemic, an that is when spelling does not reflect phonemic changes due to sandhi. Sanskrit is exceptional, in that when two phonemes are neutralized in sentential context, the writing reflects that change (so, /...as#m.../ is realized as [o#m] and is spelled that way too). Otherwise, when languages have external sandhi, those rules are typically not indicated in the spelling (for instance, there is no change in the spelling of word-final obstruents in Russian depending on whether the next word begins with a voiced consonant versus a voiceless consonant). Accordingly, you might want to subdivide the question about phonemicity into "with respect to citation forms" versus "in all contexts", since otherwise failure to reflect sentential phonology would disqualify a language. I believe that Telugu would be disqualified on that basis, since there is a process deleting vowels between certain consonants, described by Krishnamurti 1957, for instance gula:bi mogga → *gula:b mogga "rose bud". It may be that this is actually reflected in the spelling: if not, that would be an instance where the spelling includes something that is not in the phonemic output.
The sociolinguistic issues mentioned in the Telugu-informed answers point to a further way in which it is unlikely that any language has a 100% phonemic spelling. As far as know, in no language is it the rule that you "spell it as you pronounce it". Apparently, there are very many things in Telugu spelling that are not pronounced that way by all speakers. For instance there is apparently an aspiration difference in writing, which is not pronounced by some speakers. V. Sastry's 1987 SOAS dissertation mentions a number of regional and social variations, though the dissertation is not about the spelling system. From what I understand, spelling is fairly standard, and the spelling of duhkham "grief" does not change to dukkaw̃ when you pronounce the word that way.
The issues then would be whether there are letter distinctions that are not based on pronunciation (e.g. you have to know the etymology of a word), or (less likely) are there phonemic distinctions that are just ignored (e.g. tone in virtually all tone languages; vowel length in many languages; anything that doesn't conveniently translate to letters of the dominant alphabet, such as breathy voicing).