Why does "signature" have a "g" sound but "sign" doesn't?

Short answer:

When the <gn> comes word-initially or word-finally, the /g/ often gets removed.

However, in word-medial position, the /g/ is sometimes pronounced when it's followed by a vowel (because it's allowed across the syllables adn the vowel splits it up into two syllables) and is not removed.

When the <gn> is followed by a vowel, the /g/ is usually pronounced except when some suffixes like -ing, -er and -able are appended. There may be lots of exceptions, however.

Examples: When the <gn> is not followed by a vowel, the /g/ is usually silent as in the following words:

• Sign → /saɪn/
• Resign → /rɪˈzaɪn/
• Impugn → /ɪmˈpjuːn/
• Malign → /məˈlaɪn/

These words do not have a vowel after the <gn>, so the /g/ is not pronounced.

Now,

• Signature → /ˈsɪɡ.nə.tʃə/
• Resignation → /ˌrez.ɪɡˈneɪ.ʃ(ə)n/
• Pugnacious → /pʌɡˈneɪ.ʃəs/
• Malignant → /məˈlɪɡ.nənt/

These words have the /g/ because the following vowel splits up the /gn/ and makes it two syllables; /g/ moves to the preceding syllable while the /n/ moves to the next syllable.

However, some suffixes do not let the /g/ to be pronounced (I don't know the reason).

• Signable → /saɪnəb(ə)l/
• Signing → /saɪnɪŋ/
• Aligning → /əlaɪnɪŋ/ etc don't have the /g/.

Explanation:

The reason boils down to English Phonotactics (that deals with restrictions in a language on the permissible combinations of phonemes).

English Phonotactics does not permit a plosive followed by a nasal. So we cannot have an onset (beginning of a syllable) or a coda (ending of a syllable) consisting of PLOSIVE + NASAL.

Therefore, we don't have clusters like /pn/, /tn/, /kn/, /bn/, /dn/ and /gn/ etc in English because they violate the Phonotactics constraints of English.

So when the <gn> is followed by a vowel, the vowel splits up the /gn/; the /g/ moves to the preceding syllable and the /n/ moves to the next syllable.

• Signature → /ˈsɪɡ•nə•tʃə/

I don't know the reason as to why the /gn/ doesn't split up when it's followed by certain suffixes (like -ing and -able)

Greg Brooks in his Dictionary of the British English Spelling system writes:

• In a few words with final /n/ spelt <gn> /g/ surfaces in derived or related forms: compare impugn, malign, sign with pugnacious, repugnant, malignant, assignation, designation, resignation, signal, signature (all with change of vowel phoneme) – but /g/ does not surface before inflectional suffixes, as in impugns, impugning, impugned, maligns, maligning, maligned, signs, signing, signed.
• In three words with final /m/ spelt /g/ surfaces in derived or related forms: compare paradigm, phlegm, syntagm with paradigmatic (with change of vowel phoneme), phlegmatic, syntagma(tic) – but /g/ does not surface in paradigms, phlegmy.

But it doesn't explain why the /g/ is not surfaced in those inflectional words.

Another relevant quote from An Introduction To Language by Victoria Fromkin, Robert Rodman, Nina Hyams:

Another relevant quote from Nathan (2008:82):

Nathan (2008:82) asserts that not only can segments be deleted, sometimes they can be inserted instead. There seem to be two basic reasons for insertion: preventing clusters of consonants that violate syllable structure constraints in the language, and easing transitions between segments that have multiple incompatibilities — Research Gate

Why is 'signature' not pronounced [saɪgnat͡ʃə]:

Why is 'signature' not pronounced with a long vowel in the first syllable?

It's because of a fairly common phenomenon called Trisyllabic Laxing, it is a process whereby a tense vowel in a stressed syllable is shortened if two (or more) syllables follow.

Examples:

• Sign, signature --- /saɪn/ -> /ˈsɪɡ.nə.tʃə/
• Malign, malignant --- /məˈlaɪn/ -> /məˈlɪɡ.nənt/
• Profane, profanity --- /prəˈfeɪn/ -> /prəˈfæn.ə.ti/
• Sincere, sincerity --- /sɪnˈsɪə/ -> /sɪnˈser.ə.ti/
• Impede, impediment --- /ɪmˈpiːd/ -> /ɪmˈped.ɪ.mənt/
• Divine, divinity --- /dɪˈvaɪn/ -> /dɪˈvɪn.ə.ti/

Another thing I've noticed about these words is that the words in which the /g/ is pronounced in <gn> combination have a short vowel before the <gn> and the words in which the /g/ gets removed have a long vowel/diphthong before the <gn>

Examples:

Long vowel before the <gn>:

• Sign → /sʌɪn/
• Signage → /ˈsʌɪnɪdʒ/
• Signable → /ˈsʌɪnəb(ə)l/
• Assignable → /əˈsʌɪnəb(ə)l/
• Resign → /rɪˈzʌɪn/
• Design → /dɪˈzʌɪn/

Short vowel before the <gn>:

• Signet → /ˈsɪɡnɪt/
• Signature → /ˈsɪɡnətʃə/
• Signal → /ˈsɪɡn(ə)l/
• Malignant → /məˈlɪɡnənt/

I searched their etymologies because I thought they would have different etymologies but they share the same etymologies.

Yeah, just to add a bit to @Decapitated Soul's excellent answer above, the reason that sign has that ⟨g⟩ in the first place is that it was formerly pronounced in the Latin noun signum and verb signare and all their derivatives. English spelling tending to be more etymological than phonetic (for better or worse), it's been preserved even as the sound died out.

Of course, the Latin words do break the ⟨gn⟩ cluster across two syllables, as does the uncommon English borrowing signum which continues to pronounce the ⟨g⟩. For sign, that ending disappeared as Romance languages simplified Roman cases and Middle English nixed most of them altogether as it gradually merged Old English with a bunch of new French loanwords.

A minor quibble with DS's answer is that pronouncing ⟨gn⟩ inside a single syllable isn't really banned by English phonology. People were still pronouncing the ⟨g⟩ in words like gnomon as late as the 17th century. What did change was that English slowly standardized more of ⟨g⟩'s weirdness. The initial ⟨γν⟩ in Ancient Greek wasn't pronounced /gn/ but /ŋn/. As English began to standardize 'hard g' towards being a full stop instead of an occasional nasal, though, that became too uncomfortable to bother with.

Another minor quibble is that for sign & co., it really had little to do with English's own phonology. Norman and Middle French had already dropped the /g/ from their pronunciations before it entered use in English, as shown in their variant spellings sein, seine, sine, seing.