Yes, it is permissible for one who is in debt to lead the people in prayer, if he is fit to lead the prayer, because there is no evidence to suggest that it is not allowed or it is makrooh for a debtor to lead the prayer, unless he is a debtor who is delaying payment when he is able to pay off the debt, but he is delaying it and trying to avoid it. In that case he is ruled to be a faasiq (evildoer) and there is a difference of scholarly opinion concerning the faasiq. 

The evidence that it is haraam to delay paying off debts is the words of the Prophet (peace and blessings of Allaah be upon him): “If one who can afford it delays repayment, his honour and punishment become permissible.” Narrated by al-Nasaa’i (4689), Abu Dawood (3628) and Ibn Maajah (2427); classed as hasan by al-Albaani in Saheeh Abi Dawood. 

And the Prophet (peace and blessings of Allaah be upon him) said: ““For a rich man to delay repayment is wrongdoing.” Narrated by al-Bukhaari (2287) and Muslim (1564). 

Al-Shawkaani (may Allaah have mercy on him) said: There is a difference of opinion as to whether delaying payment when one is rich is a major sin or not. The majority of scholars are of the view that it constitutes fisq (evildoing), but they differed as to whether it makes one a faasiq if done once, or if that depends on whether it is done repeatedly. And is it conditional upon the demand for repayment being made or not? 

End quote from Nayl al-Awtaar (5/282). 

Among those who regarded it as a major sin was Ibn Hajar al-Haythami in al-Zawaajir ‘an Iqtiraaf al-Kabaa’ir (1/414). 

Among the fuqaha’ were those who stated that the testimony of one who delays payment of a debt is to be rejected, as is stated in al-Sharh al-Kabeer ‘ala Matn Khaleel (4/181). 

Secondly: 

In the answer to question no. 13465 we have explained the difference of opinion among the scholars as to whether it is valid for a faasiq to lead the prayer, but they are agreed that it is makrooh to pray behind him. 

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said: The imamas are unanimously agreed that it is makrooh to pray behind a faasiq, but they differed as to whether it is valid. It was said that it is not valid, as was the view of Maalik and Ahmad according to one of the two reports narrated from him. And it was said that it is valid, as was the view of Abu Haneefah, al-Shafaa’i and the other reports narrated from them. But they did not differ concerning the fact that the faasiq should not be appointed as an imam. 

End quote from al-Fataawa al-Kubra (2/309). 

Conclusion: 

If a man is poor and cannot afford to pay off a debt, this does not detract from his religious commitment or good character, and it is not valid to use this as a reason to prevent him from leading the prayers. 

But if he is able to pay off the debt but he is delaying it with no excuse and taking too long to pay it off, then he should not be appointed as an imam to lead the Muslims in prayer. 

See also the answer to question no. 47884. 

And Allaah knows best.