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Context. Rapid variations in optical flux are seen in many quasars and all blazars. The amount of variability in different classes of active galactic nuclei has been studied extensively but many questions remain unanswered. Aims. We present the results of a long-term programme to investigate the intra-night optical variability (INOV) of powerful flat spectrum radio core-dominated quasars (CDQs), with a focus on probing the relationship of INOV to the degree of optical polarization. Methods. We observed a sample of 16 bright CDQs showing strong broad optical emission lines and consisting of both high and low optical polarization quasars (HPCDQs and LPCDQs). In this first systematic study of its kind, we employed the 104-cm Sampurnanand telescope, the 201-cm Himalayan Chandra telescope and the 200-cm IUCAA-Girawali Observatory telescope, to carry out R-band monitoring on a total of 47 nights. Using the CCD as an N-star photometer to densely monitor each quasar for a minimum duration of about 4 h per night, INOV exceeding ∼1-2 per cent could be reliably detected. Combining these INOV data with those taken from the literature, after ensuring conformity with the basic selection criteria we adopted for the 16 CDQs monitored by us, we were able to increase the sample size to 21 CDQs (12 LPCDQs and 9 HPCDQs) monitored on a total of 73 nights. Results. As the existence of a prominent flat-spectrum radio core signifies that strong relativistic beaming is present in all these CDQs, the definitions of the two sets differ primarily in fractional optical polarization, with the LPCDQs showing a very low median Pop ≈ 0.4 per cent. Our study yields an INOV duty cycle (DC) of ∼28 per cent for the LPCDQs and ∼68 percent for HPCDQs. If only strong INOV with fractional amplitude above 3 per cent is considered, the corresponding DCs are ∼7 per cent and ∼40 per cent, respectively. Conclusions. From this strong contrast between the two classes of luminous, relativistically beamed quasars, it is apparent that relativistic beaming is normally not a sufficient condition for strong INOV and a high optical polarization is the other necessary condition. Moreover, the correlation is found to persist for many years after the polarization measurements were made. Some possible implications of this result are pointed out, particularly in the context of the recently detected rapid Υ-ray variability of blazars. [ABSTRACT FROM AUTHOR]