The existence of pronunciations of "forward" with dropping of the first /r/ is mentioned in "R-Dissimilation in English", by Nancy Hall (June 21, 2007). There are also pronunciations of this word where the second /r/ drops instead (so it sounds like "forwud").
The cited source for the pronunciation with /r/ dropped in the first syllable is George Hempl's 1893 "Loss of r in English through dissimilation" (Dialect Notes 1:279–81), which Hall describes as "the first description of the phenomenon [of r-disssimilation] by a linguist [...], based on his own Southern Michigan dialect" (2).
The basic idea of dissimilation as I understand it is that it's harder to hear how many occurrences of a particular sound occur in a word than it is to hear whether the sound occurs at all. Occasionally, we see the reverse process of assimilation of r-coloring, as in "sherbe(r)t".
Hall gives the following summary:
Looking at the examples in Tables 2–4 together, we see that dissimilation is usually anticipatory: of 85 examples, there are only 8 where the last /r/ of the word deletes: paraphe(r)nalia, Purmo(r)t, cereb(r)al palsy, frat(r)icide, interp(r)et, propriet(r)ess, Trist(r)am Shandy, Gira(r)d, and forwa(r)d (forward has also been reported with deletion of the first r, as shown in Table 4). In each of these exceptions, the preceding /r/ is either stressed or intervocalic, factors which decrease the chance of deletion. (6)
Since "forward" is usually pronounced with /w/, the /r/ is not intervocalic in this word, but it does occur in a stressed syllable. Evidently, though, this doesn't eliminate the possibility of deletion for all speakers (and indeed, there are other examples of /r/ being deleted after a stressed syllable in dialectal pronunciations of words like corner that you can see in Hempl's and Hall's lists).
Hall has an interesting discussion of the quality of vowels in stressed syllable when a following /r/ has been deleted.
For some speakers, dissimilatory deletion can also occur in syllables that have primary or secondary stress, as in the words in Table 4. The greatest number of these cases involve a coda /r/ deleting after /o/ as in (4a), or /ɑ/ as in (4b), or occasionally a stressed [ər] turning to [əˑ]. The restriction to these vowels is less remarkable when we consider that there are not many vowels that can appear in the position before rC in most American dialects. [...]
According to some descriptions (Swadesh 1947, Canepari 2005), these unique allophones survive when /r/ is deleted through dissimilation, so that quarter, for example, is [kwoɾər], with a first vowel unlike that of quote [kwoʊt]. This would mean that American /r/- dissimilation creates a new [o]/[oʊ] vowel contrast, despite claims in the literature that dissimilation is universally structure-preserving (Ohala 1993, Kiparsky 1995). This problem is discussed further below in section 3.5. (4-5)
In "forward", I would speculate that a pronunciation with /oʊw/ could perhaps be attributed partly to assimilation of the /r/ to the following /w/; this seems to me to happen with another consonant, /l/, in the word "always" (which I think I often pronounce roughly as "awweys" [ɑwez~ɑwɛz~ɑwɪz], even though I don't vocalize /l/ to /w/ in general).