Should he submit a false report about his travelling expenses in order to pay off his taxes?

Praise be to Allah.

Firstly: 

What is required of the Muslim is to be honest and trustworthy, so he should not tell lies or cheat, because of the sin and bad consequences involved in lying and cheating. 

Allaah says (interpretation of the meaning): 

“O you who believe! Fear Allaah, and be with those who are true (in words and deeds)”

[al-Tawbah 9:119] 

“O you who believe! Betray not Allaah and His Messenger, nor betray knowingly your Amaanaat (things entrusted to you, and all the duties which Allaah has ordained for you)”

[al-Anfaal 8:29] 

“Verily, Allaah commands that you should render back the trusts to those, to whom they are due”

[al-Nisa’ 4:58]

And the Prophet (peace and blessings of Allaah be upon him) said: “I enjoin you to be truthful, for truthfulness leads to righteousness and righteousness leads to Paradise. A man may continue to tell the truth and endeavour to be truthful until he is recorded with Allaah as a speaker of truth. And beware of lying, for lying leads to wickedness and wickedness leads to Hell. A man may continue to tell lies and endeavour to tell lies, until he is recorded with Allaah as a liar.” Narrated by al-Bukhaari(6094) and Muslim (2607).  

And he (peace and blessings of Allaah be upon him) said: “The signs of the hypocrite are three: when he speaks he lies, when he makes a promise he breaks it, and when he is entrusted with something he betrays that trust.” Narrated by al-Bukhaari (33) and Muslim (59). 

Secondly: 

If the company gives you some money to use when travelling, and asks you to report your expenses, then it is not permissible for you to lie when writing this report, whether you take the money to pay off taxes or anything else. 

That is because the sin of haraam taxes is only borne by the one who collects them and imposes them; the company has nothing to do with that, and it is not permissible to transgress against its wealth, because it is not permissible to take the wealth of a Muslim unless he gives it willingly. This is in addition to the fact that this action involves lying, so in this case it is a combination of consuming wealth unlawfully, lying and treachery – all of which are haraam. 

If you can come to an agreement with the company that you will take some money and have the right to dispose of it as you wish, then this wealth will be your property, and you can pay off taxes with it or use it for food and transportation. 

And Allaah knows best.