Phonemically1 -ing is always /ɪŋ/. The vowel phoneme2 is decided by linguists to be /ɪ/, though it can be realised in many different ways.
Phonetically3, however, it's realised as [iŋ] in some dialects of English (particularly American); that is to say, the vowel [ɪ] raises to [i] due to the effect of the following velar nasal (nasalisation).
According to The Origins and Development of the English Language by John Algeo (p26):
[I]n the South [USA], the vowels [ɪ] and [ɛ], although distinguished in most environments (such as pit and pet), have merged before nasals. Thus pin and pen are homophones for many Southerners, as are tin and ten, Jim and gem, and ping and the first syllable of penguin. The sound used in the nasal environment is usually [ɪ], though before [ŋ] it may approach [i]. [Emphasis mine]
1. /phonemic transcriptions/ are language specific transcriptions i.e. the way dictionaries transcribe words. /They/ can have [many different realisations, depending on the speaker and accent]
2. ‘A phoneme is a mental image of all the various realisations of
one and the same sound.’ (Donka Minkova) For example, the phoneme /t/ is a mental image of many realisations such as [t], [tʰ], [t̚], [ʔ] etc., in some dialects of English. By contrast, if you substituted say b for t it would change the meaning (cf. tall and ball) so we would say that /b/ and /t/ are two distinct ‘phonemes’ in English.
3. [phonetic transcriptions] transcribe actual speech sounds i.e how people speak
Best Answer
TLDR: we only write contrastive sounds (phonemes) in /slashes/. For instance, [t] and [k] contrast in English as in /tæp/ and /kæp/. [p] and [pʰ] don't contrast in English so we don't write them in slashes. Dictionaries use phonemic transcriptions, so they don't write pʰ. However, we write pʰ and p in slashes for languages that contrast them (like Chinese and Icelandic).
Explanation
So /slashes/ are used to enclose phonemes. A phoneme is a 'meaningful unit' in that it distinguishes one word from another, for example, tap is distinguished from cap by a single phoneme /t/, so /t/ and /k/ are different phonemes because tap and cap are completely different words having different meanings and we would enclose t and k in slashes becuase they contrast in English. There's a language called Samoan where [t] and [k] don't distinguish words so both [t] and [k] are the allophones ('realisations') of the same phoneme in Samoan.
Now there are other languages that have phonemic aspiration (i.e. aspiration can distinguish words) such as Icelandic and Chinese. In both those languages, [p] and [pʰ] are contrastive. For example, [pʰaːr̥] in Icelandic means 'pair' and [paːr̥] means 'bar', so their phonemic transcriptions would be /pʰaːr̥/ and /paːr̥/ respectively. However, they would mean the same thing in English because English has no phonemic aspiration, so we wouldn't write [p] and [pʰ] in slashes for English. Native speakers think of both the aspirated and unaspirated p as the same sound (the same phoneme /p/).
Likewise, [tʰ], [t], [ɾ], [t̚] and [ʔ] occur as allophones of the phoneme /t/ in English. Although native speakers do (and can) hear the difference between all those sounds, they think of them as the same sound—/t/.