You put forward two sentences:
- I have to be sharp and clever lest I go hungry? (infinitive)
- John and his friends feared lest the inquiry promised into the extent of the hated forest areas would be carried out too rigorously? (past tense?).
You asked: "Are both examples acceptable?"
The answer is "no". "Lest...would" is not a standard construction. The verb should be either the subjunctive form (which isn't the same thing as an infinitive, but has the same form as the bare infinitive), as in your first example, or it should be a phrase of the form "should"+bare infinitive, e.g. "should go".
The two forms ("lest I go", "lest I should go") are interchangeable. Regardless of the tense of the verb in the main clause, both forms remain available and remain interchangeable.
"The idiomatic construction after 'lest' is 'should', or in exalted style the pure subjunctive; good writers rarely use 'shall', 'may' and 'might'... 'Will' and 'would' are merely a special form of the inability to distinguish between 'shall' and 'will'" (Fowler's Modern English Usage, 1965). While Fowler is sometimes too prescriptive, his advice here fits with that in other references in such as A Practical English Grammar (Oxford University Press, 1986), which recommends "should" here and does not mention the possibility of "would" or any other auxiliary. The Oxford English Dictionary has a "lest... might" from the great Scottish poet Robert Burns (1795), but all its more recent examples have either "should" or the subjunctive.