TLDR

The pronunciation of 'iron' in standard varieties of English is EYE-URN (BrE: /'aɪən/, AmE: /'aɪrn/) and not EYE-RUN (which is also a common pronunciation of 'iron' in some varieties of English) because of a very common process called Metathesis. It's defined as the transposition/rearrangement of letters, syllables or phonemes (sounds) in a word.

Explanation

There are some other words that show the same change; 'wasp' used to be waps, 'bird' used to be brid and 'horse' used to be hros, but they've changed over time. Why is that?

It's because of a very common process called Metathesis. /'aɪən/ is the metathesised version of (/'aɪrən/). 'Iron' commonly used to be pronounced the way it's spelt (/'aɪrən/), but due to metathesis, its pronunciation became EYE-URN (/'aɪən/). Other words such as horse, bird, third etc., reflect the change in spelling; however, 'iron' doesn't reflect that change probably because metathesis applied to it after the spelling was standardised.

Metathesis:

Metathesis is defined as ’the transposition/rearrangement of letters, syllables or phonemes (sounds) in a word’.

Examples:

• 'wasp' used to be waps (wæps) [transposition of p and s]
• 'bird' used to be brid [transposition of i and r]

{Historical Metathesis}

• 'mix' being pronounced /mɪsk/ rather than /mɪks/ [transposition of k and s]

• 'desk' being pronounced /dɛks/ rather than /dɛsk/ [transposition of k and s]

• 'modern' being pronounced /'mɒdɹən/ instead of /ˈmɒd(ə)n/ (US: /ˈmɑː.dɚn/)

• 'pattern' being pronounced /pætɹən/ instead of /ˈpæt.ən/ (US: /ˈpæɾ.ɚn/)

• Another famous example from Shakespeare's 'The Tempest' is the figure of Caliban whose name originates from a phonological metathesis of /n/ and /l/ in 'cannibal'. _[ThoughtCo]

/r/-metathesis

Words having /r/ + vowel sequences are more susceptible to metathesis than others. According to A grammar of Old English Phonology by Richard Hogg, ‘R-metathesis normally occurs when /r/ is followed by a short vowel and a dental or alveolar consonant, usually /n/ or /s/’.

'Iron' is an example of /r/-metathesis. It was probably pronounced /'aɪrən/ (EYE-RUN) at one point, but it got metathesised to /'aɪərn/ (EYE-URN). However, the spelling remained unaffected.

Other examples of metathesis of /r/ include:

• bird from brid (bridde)
• third from thridde (ðridde)
• horse from hros

The silent R in 'iron' in BrE

The reason why the r in 'iron' is absent in British English is because the r is followed by a consonant now (followed by /n/ in /'aɪərn/) and British English is non-rhotic, meaning the r is only pronounced when followed by a vowel. The same thing happened to 'bird', 'horse' and 'third' too (i.e. the r is followed by a consonant, so it's silent).

There are different types of metathesis, 'colonel' (pronounced KE(R)-NUHL /'kɜː(r)nl̩/) can also be said to be a product of metathesis. (See this answer on ELU for the spelling and pronunciation of 'colonel')

According to Wikipedia, the reason for ‘common speech errors’ is also metathesis.
Examples include:

• perscription for prescription
• interduce for introduce
• revelant for relevant
• foilage and foliage

References:

• A Dictionary of Phonetics and Phonology - Trask and Lawrence
• English Words: History and Structure by Stockwell and Minkova
• A Grammar of Old English Phonology by Richard Hogg
• Metathesis - ThoughtCo
• The Sounds of Language: An Introduction to Phonetics and Phonology - Elizabeth C. Zsiga

TLDR

The W in 'two' and 'sword' is silent because of a sound change that took place somewhere between Old English & Middle English. The change applied to words in which the W was preceded by [s, t] and followed by a back vowel like [ɔ o ɑ u] etc.

'Swore' and 'sworn' also lost their W's at one point, but were later on restored by analogy with swear.

The silent W in 'sword' and 'two'

Between Old English and Middle English, there was a sound change through which a /w/ was lost in the environment of a preceding [s] or [t] (and sometimes [h]) and a following back vowel ([ɒ ɔ o ɑ u] etc). So 'two' was twā and its pronunciation was /twɑː/ in Early Middle English, it became /twoː/ and then it lost the /w/ and became /toː/ (the vowel /oː/ was shifted to /uː/ because of the Great Vowel Shift).

'Sword' was pronounced /sword/, it became /sord/ (or /sɔrd/). The conjunction/adverb 'so' used to be pronounced with a /w/, but it lost its /w/ due to the same reason. The change also affected 'who' as it was hwa before.

However, the change didn't occur when the /w/ was followed by a front vowel; for instance, 'twin' and 'swift' didn't lose their /w/'s because the w's in those words were followed by a front vowel /ɪ ~ i/.

Now 'swore' and 'swollen' deviate from this rule. They do have back vowels after the /w/ and there is an /s/ before the /w/, yet the /w/ is preserved. That's because of Analogical change.

Analogical change

Before explaining 'swore', I'd like to elucidate 'analogy' (or analogical change). It's ‘a type of language change in which some forms are deliberately changed merely to make them look more like other forms’ [Trask]. Larry Trask, a phenomenal linguist, gives a remarkable insight into 'analogy' in his book Trask's Historical Linguistics. I'll just explain it the way Trask has explained with another example.

Suppose I tell you that frumicate is a rare and obscure English verb meaning to put on airs. I'm fairly certain you've never heard it before (because you haven't, have you?). What do you suppose its past tense is? You've never heard its past tense before, but I'm pretty sure you'll say frumicated (agree?). How do you know its past tense is frumicated?

‘You do it by invoking analogy’—that is, you assume that the required past tense is formed according to a very common pattern that is already familiar to you from large numbers of other English verbs. In this case, the pattern for forming past tense is so widespread and regular that it actually constitutes a rule of English grammar for regular verbs. ^{[Adapted from Trask]}

Trask gives another example:

• teach → taught
• catch → ?

The past tense of 'catch' was formerly catched, but as a result of this analogy, it has become caught (which is now the standard past tense of 'catch').

The /w/ in 'swore' and analogy with 'swear'

The same change (loss of /w/) should've happened with 'swore' and 'swollen', but these are nonetheless pronounced with a /w/ today.

It's possible though that the analogy of these forms—which always retained their /w/s because they had front vowels—prevented the regular sound change from affecting 'swore' and 'swollen'.

Or they did actually lose their /w/s at one point, but were later restored by the analogy with swear and swell. [see Analogy and levelling — Trask's Historical Linguistics pp101-102]

Wikipedia says the same: /w/ in swore is due to analogy with swear. Another Wikipedia article says that ‘[a]n example of analogical maintenance would be the perseverance of /w/ in swollen by analogy with the present tense swell (contrast with sword, where the /w/ is lost by regular sound change)’.

The change didn't apply to swear and swell because the /w/ in those words was followed by a front vowel and the sound change didn't affect words in which the /w/ was followed by front vowels.