There are some explanations on what YubiKey does here. Basically, the password which the YubiKey "types" (from the point of view of the computer, it is a keyboard) can be either a static password, or a one-time password. If it is a static password, then you just revealed it, and it is time to be very sorry (and promptly change that password).
The one-time passwords, what YubiKey produces follows HOTP. The cryptography in HOTP is such that it is not computationally feasible to recompute the "master secret" from one or several one-time passwords produced with HOTP. Moreover, each password is internally computed from a counter. The YubiKey and the server both maintain the same counter, and the server allows for some limited lack of synchronization. Namely, when the server's current counter has value n and receives a password as authentication attempt, it will internally generate the passwords for values n+1, n+2,... up to, say, n+100 (that's configurable). If a match is found with (say) password n+17, then access is granted and the server's counter is set to n+17; otherwise, connection is rejected and the server's counter is not changed.
Therefore, what you inadvertently published "on the Internet" is a password which will grant access to the corresponding server, until your own next authentication on that server, because that authentication will update the server's counter to a further counter value. In a way, using OTP with counter value k invalidates all OTP values with values j < k. Which leads to the following recovery procedure: if you published an OTP value, quickly connect to the server so as to invalidate that published value. Afterwards, you can just ignore it; once invalidated, it is harmless.
(Note: if you repeatedly generate a lot of "blank" passwords with your key without authenticating to the server, your YubiKey may go out of synch with that of the server -- the key using counter values way beyond what the server would currently accept. Don't let your 3-year-old play with your YubiKey ! In a similar situation, for infrared car keys, counter synchronization is forced through RFID when you start the engine.)