This may depend on the person, or on how you choose to analyze English syllabification (which is a dubious and contentious matter). I believe there are some people who argue that "lemon" should be split up as /ˈlɛ.mən/ and "delicious" as /dɪˈlɪ.ʃəs/, because of general principles of syllabification such as avoiding onsetless syllables or maximizing the onset.
In the specific case of the word "tailor," I feel that I pronounce it, at least on some level, as /ˈteɪl.ər/, because I smooth the phoneme /eɪ/, which is usually a rising diphthong, to a monophthong or perhaps a falling dipthong: [ˈteɫɚ] or [ˈteə̯ɫɚ]. This smoothing does not occur for me in words like playlist, grayly, or rayless. The easiest way I see to explain this contrast is by different syllabification of the intervocalic /l/ in these words: /ˈteɪl.ər/ vs. /ˈgreɪ.li/. But other analyses are possible.*
There are other types of allophony that may occur in other accents exclusively before tautosyllabic /l/.
John Wells has a post about vowel backing in British English accents that occurs in this position: "newly minimal". He says
Christian Uffmann reports an interesting development that he has observed in 18-year-old applicants for university places at Sussex. He asked them whether ruler (king) and ruler (for measuring or drawing straight lines) are homonyms or not. The unanimous opinion was that they are not homonyms, because they are pronounced differently. (So technically they would be homographs.)
This is because in southeastern England /uː/ has developed two very distinct allophones: a truly back [uː] before tautosyllabic (or stem-final) /l/, but a fronted quality approaching [yː] in other positions. The kingly ruler, ˈruːlə, is taken as transparently bimorphemic, rule#(e)r, so retains the back uː of rule; but the measuring ruler, ˈryːlə, has lost touch with its origins and is taken as an unanalysable unit, with a corresponding clear l and fronted vowel yː.
This means that there might be variation between speakers in the pronunciation of "tailor" depending on whether they analyze it as a morphologically simple or morphologically complex. Etymologically, the word did come into English as a whole from French taillour/eur, but its status as a noun of occupation may increase the likelihood that speakers analyze the final -or as a synchronic suffix (even though the base "tail" is not ususally used as a corresponding verb). I don't know if there are any speakers who don't rhyme "tailor" with "nailer" (one who nails), but it seems at least possible.
*The contrast for me between "tailor" and "grayly" doesn't have to be explained in terms of syllabification
As I said, to me it feels like the first syllable of "tailor" is "tail", with coda /l/. However, an alternative analysis could be that in my accent, vowel smoothing and l-"darkening" are word-level processes and /l/ is darkened when it is in non-foot-initial position at the word level, regardless of whether it is in the syllable coda or the syllable onset (for more elaboration of this kind of analysis, see "An amphichronic approach to English syllabification", Ricardo Bermúdez-Otero, 2013). Compare t-lenition, which does not occur foot-initially at the word level, but which does apply foot-medially, not only after "short" vowels but also after "long" or "strong" vowels—as in the word "waiting" [weɪt̬ɪŋ], which would be syllabified according to the rule that you have learned as /weɪ.tɪŋ/.