What did the scholar Ibn 'Uthaymeen mean by saying that writing the verse "{And say, 'Do [as you will], for Allah will see your deeds, and [so will] His Messenger'}" on some works, after the death of the Messenger (peace be upon him), is considered an error, ignorance, and a lie against him?

What the Sheikh said is correct and his intention is clear. He negates the Prophet, peace and blessings be upon him, seeing deeds in this world after his death, and he does not intend to negate the other meanings included in the verse, such as the presentation of deeds to Allah, His Messenger, and the believers on the Day of Resurrection. The verse is a warning from Allah Almighty to those who violate His commands, that their deeds will be presented to Him, and to the Messenger and the believers on the Day of Resurrection, and this may become apparent to people in this world. The vision here encompasses both knowledge and sight. The believer must take the means, fulfill what Allah has obligated him to do, and abandon what Allah has forbidden him to do.