I found this succinct examination of Chirrut. It seems to me to underline that he maybe no Jedi but he is a force user
He's not outright stated to be Force sensitive, but he sure does things that can lead no other conclusion. Quacks, walks, and swims like a duck Force Sensitive.
All quotes below are from Alexander Freed's official novelization:
Aside from his fighting abilities (See Terriblefan's awesome answer) he:
Sensed that Jyn wore a Kyber crystal necklace (while it was hiddent beneath her clothing)
“Would you trade that necklace for a glimpse into your future?” The voice of the pilgrim. Jyn frowned and took another step forward, trying to locate the source.
“Your necklace?” the man repeated.
Jyn felt the crystal against her skin. Her necklace was still hidden, buried under layers of cloth.
And the man was blind.
“I am Chirrut Îmwe,” the man said.
“How did you know I was wearing a necklace?” Jyn asked, and felt like a fool, like a mark, even as she spoke.
It was the reply of a con artist. Jyn shook her attention from Chirrut to search for his partner (he must have had a partner, one who had spotted her necklace somehow) and immediately found her quarry: a hulk of a man with hair as wild as Chirrut’s was neat, in a filthy civilian flight suit and battered red plastoid armor half concealed under a wearable tarp. On “How did you know I was wearing a necklace?” Jyn asked the second man, who shook his head slowly and snorted. ... (Chapter 4)
Not only that, but he knew it was a Kyber crystal shard!!!
“What do you know of kyber crystals?” Chirrut asked. His tone was patient, prompting.
Called Cassian "Captain" when he had no way of knowing his rank
“There is more than one sort of prison, Captain,” Chirrut said. “I sense that you carry yours wherever you go.”
Cassian frowned and turned back to the lock and the cavern. It was some minutes later that he realized no one had told Chirrut he was a captain. (Chapter 5)
Spotted someone (ended up Bodhi) in the next cell. Blind. NEITHER Cassian nor Baze noticed anyone was there.
“Who’s the one in the next cell?” Cassian tore his eyes from the guards and glanced over to Chirrut. It was the first time the blind man had spoken for nearly an hour. ... Baze grunted and shuffled to his feet. “What? Where?” He crossed the alcove, lightly shouldering Cassian aside to make room at the door. He peered into the darkness of the cell across the way; all Cassian could see was shadows, but Baze pulled back abruptly, snarling. “An Imperial pilot.”
Realized that earth shaking while in the cell wasn't Proton Bombs but something worse (Death Star blast)
“Proton bombs,” Baze said, turning his eyes to the ceiling.
Chirrut shook his head. “No.” But he ventured no alternatives. (Chapter 8)
Claims to be able to sense when someone has dark intent
She should have been irritated by the curt explanation. Instead she could muster only vague confusion. “Why do you ask that?” she said, looking to Chirrut now. “What do you mean, Does he look like a killer?”
“The Force moves darkly near a creature that’s about to kill,” Chirrut answered. (Chapter 10)
Claims to sense anger in Baze (though that could just be knowing his close friend)
Baze readied himself to pull the trigger on his cannon. To burn the platform with more blaster bolts than there were drops of rain on Eadu. “I sense anger in you,” Chirrut said.
Baze himself, internally in his head, claims that the Force helps Chirrut
They hunted in sync, Chirrut always prowling near the rebels and Baze always prowling near Chirrut. Baze did not limit his targets to those who might spot the blind man, but he kept Chirrut under observation nonetheless; where the Force would fail Chirrut, Baze would not.
He himself doesn't outright deny it, and claims the Force protects him, whatever that means:
Cassian, evidently, had other things on his mind. He watched Chirrut’s partner. “Is he Jedi?” he asked, with the hushed doubt of a man on the verge of a great discovery.
“No Jedi anymore,” Chirrut’s partner said. “Only dreamers like this fool.” Chirrut shrugged mildly. “The Force did protect me.”