Tokens (like tracking) only work when the email is sent from Marketo itself. Reason being, it queries fields in your Marketo database on send in order to pull them through into the email.
If the other company is sending an email to recipients from another system, that system might have its own 'tokens', alternatively you keep the text generic.
Thanks- good to know. Is there any way to take an email I have created using tokens and make sure it is safe for another party to send without having to rebuild the entire email?
If you used the Marketo Email Editor to create the email and exported to html, you should really only need to remove the tokens (replace with their tokens/generic text).
Because they are sending it from another system, I'd suggest that you request for them to send you a test from their system in advance of the blast to ensure everything renders as you are expecting.
You'll also need to have them set up the From & To Address, Subject Line, Click to View Online and Unsubscribe links on their end. As a side note the Unsubscribe should technically be linked to THEIR unsubscribe page, since they are the senders.
You might want to confirm their sending on your behalf is spam compliant in your country, as I know this is a questionable practice in Australia.