Which interrupts are enabled when programming an ATTiny85 with Arduino IDE?
The only interrupt that is always running is the one which generates the count for millis() and micros(). Disabling interrupts disables your ability to keep track of time and use delay().

Timer interrupts are also used by the tone() function and for sending / receiving serial data if the chip has a UART.

Disabling interrupts would also affect the use of external interrupts and pin change interrupts should you be using them.

It sounds, from your brief description, that you would only be effecting the use of millis() and delay() since you aren't using the other systems that use interrupts at all.