The faxes still come through serially with the receiving machine able to set the maximum rate of transmission. With one phone line and four minutes per fax that's a maximum of 360 requests per day. Send some long outgoing faxes and availability for receiving notices goes down.
In any event, spooling up a lot of faxes would tie up YC's fax machine, not YC's staff.
Couldn't you just use some online faxing service? That would prevent the 360 request/day limit you mentioned. I don't think any place sending mass DMCA take downs would be sending faxes themselves.