Turning off some legends in a ggplot

You can use guide = "none" in scale_..._...() to suppress legend.

For your example you should use scale_colour_continuous() because length is continuous variable (not discrete).

(p3 <- ggplot(mov, aes(year, rating, colour = length, shape = mpaa)) +
   scale_colour_continuous(guide = "none") +
   geom_point()
)

Or using function guides() you should set "none" for that element/aesthetic that you don't want to appear as legend, for example, fill, shape, colour.

p0 <- ggplot(mov, aes(year, rating, colour = length, shape = mpaa)) +
  geom_point()    
p0+guides(colour = "none")

UPDATE

Both provided solutions work in new ggplot2 version 3.3.5 but movies dataset is no longer present in this library. Instead you have to use new package ggplot2movies to check those solutions.

library(ggplot2movies)
data(movies)
mov <- subset(movies, length != "")

You can simply add show.legend=FALSE to geom to suppress the corresponding legend