> What type of attacks are there that do not use open TCP or open UDP ports?

This is way too general of a question. I'm answering this very literally, not to be a jerk, but because in security it's best to assume nothing.  Here are some classes of attacks that do not use open TCP or UDP ports:

* Social engineering: get someone to connect outbound from the machine to an attack site (or attach a thumb drive or bad media)
* Physical access, keyloggers, etc
* An attack at the IP level (IP stack vulnerabilities)
* NTP: usually turned on by default and might not be bug-free
* DHCP:  is dhcp turned off?  an attacker on the local net could push a PXE boot image to your ethernet card and load their OS.

> Is it safe to assume that no open ports means no remote access?

 * Drive-by downloads that get executed could call out (including pinging out bound and getting ctrl data via ping reply)
 * Package managers could pull down trojaned software setting up call-out malware
 * Bluetooth access

I think your real question should be:  "What kinds of remote exploits can be used to root my machine if it has no open TCP or UDP ports?"  

*Attacks* can mean any number of things to people including Van Eck Phreaking.