The correct option is
A live
The verbs that do not explain the tense of the sentence and do not agree with the subject are called non-finite verbs. Non-finite verbs are usually infinitives, gerunds and participles.NOTE: As the sentence is in the present tense, we will choose the non-finite verb accordingly.
Option B - As 'can't' is a helping verb, we need a main verb to support it. 'To live' is an infinitive and it can't be used here as the verb is not in its original form. Hence, option B is incorrect.
Option C - Though 'living' can be a non-finite verb, we can't use it here as after 'can't/can', the verb is always in its original form. Hence, option C is incorrect.
Option D - 'Lived' is the past form of the verb. Hence, it is definitely can't be used as the sentence is in the present tense. Thus option D is incorrect.
Option A - 'Live' can be used after 'can't' as it is grammatically correct. Also, 'live' doesn't explain the tense of the sentence. Hence, it is a non-finite verb here. Thus option A is the correct answer.
'He can't live with her.'