They likely won't fail immediately, but if your front tire blows out it's very like you'll immediately crash. I rode my share of 30-year-old 27" tires in college and never crashed because they failed, but it's penny-wise, pound-foolish, because blowing out a front tire could result in your face making very close acquaintance with the NYC pavement you named yourself after.

A basic pair of 27" tires is $30 at Recycle-a-Bicycle or $20 online:
http://www.amazon.com/Cheng-Shin-C63...rds=27%22+tire

The 30-year-old brake pads also won't do a very good job of stopping you. A set of salmon Kool-Stop Continentals will do much better:
http://www.amazon.com/Kool-Stop-Bicy...op+continental
Only in front is still better if you're on a really tight budget.