The town was only liberated from IS last year, with many only having returned recently to rebuild their homes.
However, violence broke out again last week, meaning many Christians in the town have been displaced to nearby Alqosh and surrounding areas.
The conflict is between Peshmerga Kurdish forces and Iraqi forces.
The Iraqi Christian Relief Council reported on Twitter that: "Kurdish Peshmerga have been shooting indiscriminately...to make it appear as though the Iraqi Army is injuring civilians".
They added: "Three Christian children were caught in the crossfire and injured while playing soccer. One woman was also injured."
The town's inhabitants were told by a representative of the Iraqi government that they had "until sunrise to evacuate", according to Fox News.
The Iraqi army were planning to forcibly remove Peshmerga from Telskuf - which is 19 miles north of Mosul in the morning.
It seemed they warned the town civilians as they wanted them out of the way.
Telskuf was freed from IS control in 2016 and rebuilt with $2 million (£151 milliion) of aid money from Hungary.