に堪えない【にたえない】

に堪えない has two meanings, and what goes in front decides which. After a noun for a feeling – かんしゃ感謝, どうじょう同情, いかん遺憾 – it means the feeling is more than you can hold in: "deeply", "beyond words". After the dictionary form of みる見る, きく聞く or よむ読む, it means something is too awful to watch, hear or read.

The verb is in the dictionary form, not the ます-stem.

• × 見に堪えない → 見るに堪えない

Feelings

The feeling belongs to the speaker, so this use mostly appears in the first person, in speeches, letters and formal apologies. The set of nouns is small: 感謝, 同情, 遺憾, かんげき感激, つうこん痛恨.

Too awful to look at

The second use judges the thing, not the speaker's feeling. It often comes before a noun or with ほど.

Register

Both uses are formal and written. In conversation you say ほんとうに本当にありがとうございます for the first and みていられない見ていられない for the second.