わけにはいかない is a statement that combines わけ ‘to be so’, ‘reason’, or ‘cause’, with the 格助詞（かくじょし） (case marking particle) に, the 副助詞（ふくじょし） (adverbial particle) は, and the ‘cannot go’ meaning of 行（い）かない. It can be translated as ‘cannot afford to (A)’, or ‘impossible to (A)’. However, the literal meaning is much closer to ‘it cannot be so that (A)’.
This structure will be used directly after the non-past form of verbs, in order to highlight them as being either highly undesirable, or highly unattainable.

• この料理（りょうり）は妻（つま）が頑張（がんば）って作（つ）ってくれたやつだから不味（まず）くても食（た）べないわけにはいかない。
  My wife made this food for me, so no matter how bad it tastes, it cannot be so that I won’t eat it.
• 明日（あした）は人手（ひとで）が足（た）りないから、休（やす）むわけにはいかない。
  Since we are short-handed tomorrow, I cannot afford to take a day off.

Although わけにはいかない primarily highlights things that are undesirable, it may also highlight something that is desirable, but impossible for some reason. This will depend largely on the speaker’s tone of voice, and context of the conversation.

• このジャケットが欲（ほ）しいけど、今月（こんげつ）はお金（かね）を使（つか）い過（す）ぎたから買（か）うわけにはいかない。
  I want this Jacket, but because I spent too much money this month, it cannot be so that I buy it.
• 先輩（せんぱい）に誘（さそ）われたから、断（ことわ）るわけにはいかない。
  I was invited by my senpai, so there is no way that I can decline the offer.

In these types of sentences, the expression is quite similar to the ‘I guess it’s just not meant to be so’ nuance that can be conveyed in English, highlighting regret.