The Muslim army can either be fighting against other Muslims or against the disbelievers. In the first case it is not permitted to seek the help of non-Muslims.
In the second case, i.e. in wars against disbelievers, there is a difference of opinion.
A group of the Ulema consider it impermissible based on e.g. the refusal of the Prophet ﷺ to accept the help of a non-Muslim in the battle of Badr:
فارجع فلن أستعين بمشرك
Go back, I will not seek help from a Mushrik
— Sahih Muslim
The majority (Hanafi, Hanbali, Shafi'i) consider it permissible when there is a need for it and as long as certain conditions are fulfilled.
Their evidence for its permission includes:
Safwan bin Umayyah helped in the battle of Hunain while he was still a disbeliever.
فخرج رسول الله صلى الله عليه وسلم قبل هوازن بحنين . فأرسل إلى صفوان بن أمية يستعيره أداة وسلاحا عنده ... ثم خرج صفوان مع رسول الله صلى الله عليه وسلم وهو كافر فشهد حنينا والطائف وهو كافر
The Messenger of Allah, may Allah bless him and grant him peace, went out toward Hawazin at Hunayn. He sent to Safwan ibn Umayya to borrow some equipment and arms that he had. ... Then Safwan went out with the Messenger of Allah, may Allah bless him and grant him peace, while he was still a kafir. He was present at the battles of Hunayn and at-Ta'if while he was still a kafir.
— Muwatta Malik
Some Jews are reported to have fought with the Prophet ﷺ:
أن النبي صلى الله عليه وسلم أسهم لقوم من اليهود قاتلوا معه
The Prophet (ﷺ) gave a portion to some people among the Jews who fought along with him
— Jami at-Tirmidhi
There is a prophecy about fighting with the Romans:
ستصالحون الروم صلحا آمنا فتغزون أنتم وهم عدوا من ورائكم
You will make peace with the Romans in a secure truce, and you and they will fight an enemy who is behind you.
— Sunan Abi Dawud