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

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.