Great question! I think I have found one example. A few times, someone asked Jesus to heal them and, after healing, He sent them to the "as a testimony to them". Examples1 are:
Note that in both cases the petitioner started with "Lord" - implying that he was a believer.
Contrast that with this story from Mark 1:40-45:
A man with leprosy came to him and begged him on his knees, ‘If you are willing, you can make me clean.’
Jesus was indignant. He reached out his hand and touched the man. ‘I am willing,’ he said. ‘Be clean!’ Immediately the leprosy left him and he was cleansed.
Jesus sent him away at once with a strong warning: ‘See that you don’t tell this to anyone. But go, show yourself to the priest and offer the sacrifices that Moses commanded for your cleansing, as a testimony to them.’ Instead he went out and began to talk freely, spreading the news. As a result, Jesus could no longer enter a town openly but stayed outside in lonely places. Yet the people still came to him from everywhere.
Note two things:
- This man does not address Jesus as "Lord" (see Romans 10:9 and 1 Corinthians 12:3).
- Jesus gives the man specific instructions: not to tell anyone and to offer a sacrifice as a testimony to the priests. The man does neither, thereby inhibiting Jesus' ministry.
Together, these two things imply to me that the man was not a believer nor was intent on making any effort to become a believer - he simply wanted healing.
Conclusion (which perhaps includes a pertinent warning for your ministry): Jesus wanted to use this man and gave him specific instructions but the man himself disobeyed.
1 Though possibly these two Gospel accounts are about the same event.