I was reviewing my Anki cards and it said that 減少 and 増加 are antonyms. Now, for extra practice, I try to give at least one kunyomi reading to each kanji. Then it occured to me that ふえる and へる sounded similar. This question may sound crazy and naive, but could they be etymologically related, considering that some words spelled with え now used to be spelled with へ?
In Japanese, phonetic equivalence is generally a very poor predictor of common etymology. Japanese is well known as having a preponderance of homophones, mostly due to the accretion of possible readings for each kanji which has developed over the centuries. But even among wago (和語), verbs ending in 'eru' are extremely common. Still, it is always worth researching individual cases.
A search of the words on Daijirin doesn't reveal evidence of a common etymology, at least from the definitions here and here. I also checked Kojien (広辞苑第六版) - it lists 減る as appearing in the 'Genpei Sesuiki', an extended version of the Heike Monogatari (approximately 12th century). According to Daijirin, 減る also appears in the Koyo Gunkan (17th century), so presumably, the word was in regular use during the interim period. I couldn't find any contemporaneous examples of 増える, checking in the same Kojien edition and Shinmeikai (新明解国語辞典 第五版). That doesn't mean it wasn't in regular use, of course. But without solid examples of usage that provide context for the words, I can't find any evidence to support your hypothesis of a common etymology.
Nonetheless, it's an interesting hypothesis in light of the phonetic changes you mention. However, I think we need more details on how those specific words might have undergone change before we draw any solid conclusions about their derivations.
My main reference is the 日本国語大辞典【にほんこくごだいじてん】 (NKD). This is one of the better monolingual Japanese dictionaries for providing historical details and etymologies.
Digging further, we find that this ～ゆ ending might derive from the Old Japanese auxiliary or verb ending that indicates spontaneous action that happens on its own, as well as potential and passive senses. This underlying meaning of "happens on its own" agrees with the fact that this verb is intransitive.
- We can find a hint about the lurking "y" in the verb ふえる from its transitive counterpart ふやす. This ふやす can also be analyzed as the causative form of ふゆ: verb root fuy- + linking vowel -a- + causative / transitive ending -su.
The ～ゆ suffix had the so-called 下二段活用【しもにだんかつよう】 or "lower-bigrade conjugation" pattern, where the conjugation forms ended in either -u or -e. This commonly shifted from ～ゆ to ～える as time passed, and this is the same ～ゆ that we also see in classical 見【み】ゆ and 聞【き】こゆ, modern 見【み】える and 聞【き】こえる.
The core meaning of this verb has consistently been "to increase".
Meanwhile, the NKD entry for へる shows that this has pretty much always had the form へる, since it was first documented in 1170. The verb root is her-, or in older stages of Japanese, fer-.
(See also this other post about the historical sound shifts for the はひふへほ kana.)
The core meaning of this verb has consistently been "to decrease".
- The phonology (sound) suggests independent roots: fuy- and fer- are quite distinct, and there are no explainable sound shifts that could apply to suggest that these come from some shared origin.
- The semantics (meanings) also suggest independent roots. While opposites sometimes have shared origins (consider English pairs like positive terrific and negative terrible, or awesome and awful), we can usually trace the development of such meaning pairs. For fuy- and fer-, the two have always had opposite meanings, for as long as the terms have been used in written Japanese.
Conclusion: While it is possible that these two terms are cognates that share a common ancestor root, and the sense and sound developments have simply been lost to history, in consideration of the known development of both words, it appears unlikely that these terms are related.