Typically, no. In SPI all lines are actively driven high or low rather than an open drain configuration. The bus master drives SCK (SPI clock), MOSI (master out, slave in) and CS (chip select), and the slave device(s) drive the MISO (master in, slave out) line(s). From an operating perspective, pullup/pulldown resistors are not required.
Now having said that, in some cases it's considered appropriate to add weak pullup/pulldown resistors to the data lines to ensure that they're in the expected state on power up and to prevent glitches from putting slave devices into weird states.
Virtually all SPI devices that I know of are CMOS with push-pull outputs and high-impedance inputs, which are much more friendly to work with. Until you try to drive the bus with two devices by accident. See my other comment about series resistors. I have a circuit right now that's operating a SPI bus with a 100 MHz clock on a small two-layer board and I'm getting away with it. A real engineer wouldn't allow this, but it's strictly for a research project.