We need to move quickly to close this deal.

7 [no object] : to take action : act

• We need to move quickly to close this deal.
• He was waiting for the right time to move against his enemies.
• She hasn't yet moved on their recommendations.
• We must move [=proceed] very carefully to avoid offending them.

Source: https://www.britannica.com/dictionary/Move

Is the two red "to" "in order to"?

GoldfishLord said:

7 [no object] : to take action : act

• We need to move quickly to close this deal.
• He was waiting for the right time to move against his enemies.
• We must move [=proceed] very carefully to avoid offending them.

Source: https://www.britannica.com/dictionary/Move

Is Do the two red "to"s mean "in order to"?

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Note my corrections to your question above. Using "in order to" in place of them would result in sentences that mean the same thing. That's not the same as saying that "to" means "in order to".