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The question states:
"What does a router do if it has no EIGRP feasible successor route to a destination network and the successor route to that destination network is in active state."

The answer should be:
It routes all traffic that is addressed to the destination network to the interface indicated in the routing table.

Explanation:
Since the successor route is in active state, it does not need a feasible successor route. EIGRP only cares that a route is working in this case. Now if the successor route should fail, then it would send a multicast query packet to adjacent neighbors requesting available routing paths to the destination network.

A destination in the topology table can be marked either as Passive or Active. A Passive state is a state when the router has identified the successor(s) for the destination. The destination changes to Active state when current successor no longer satisfies the Feasibility Condition and there are no feasible successors identified for that destination (i.e. no backup routes are available). The destination changes back from Active to Passive when the router received replies to all queries it has sent to its neighbors. Notice that if a successor stops satisfying the Feasibility Condition but there is at least one feasible successor available, the router will promote a feasible successor with the lowest total distance (the distance as reported by the feasible successor plus the cost of the link to this neighbor) to a new successor and the destination remains in the Passive state.

An active route is not usable unless it becomes passive again.

So at best the answer should be to pick two answers? Because it will still be able to route traffic to the destination network.

The explanation given with this question says that no traffic will be forwarded when EIGRP is in active state for a destination.

If no feasible successor is available and the successor fails, a discovery process in which the router that lost the successor sends EIGRP queries to neighbors and calculates new successor based on their replies, is used to find any alternate pathways to the lost networks. During the query process the router is said to be in Active state for that destination. This means no traffic will be forwarded to the lost destination till the query process is finished and DUAL calculates a new route.

If you believe statement is incorrect please quote your reference from cisco.com.