Close binary central stars of planetary nebulae (PNe) must have formed through a common envelope evolution during the giant phase experienced by one of the stars. Transfer of the angular momentum from the binary system to the envelope leads to the shortening of the binary separations from the radius of red giant to the radius of the order of few tenths of AU. Thus, close binary central stars of planetary nebulae are laboratories to study the common envelope phase of evolution. The close binary fraction in the Galaxy has been measured in various sky surveys, but the close binary fraction is not yet well constrained for the Magellanic Clouds, and our results may help the study of common envelope evolution in low-metallicity environments. This paper presents a continuation of our study of variability in the Magellanic Cloud planetary nebulae on the basis of data from the OGLE survey. Previously, we had analysed the OGLE data in the Small Magellanic Cloud. Here, the study is extended to the Large Magellanic Cloud (LMC). In this paper we search for close binary central stars with the aim to constrain the binary fraction and period distribution in the LMC. We identified 290 counterparts of PNe in the LMC in the I-band images from the OGLE-III and OGLE-IV surveys. However, the light curves of ten objects were not accessible in the OGLE database, and thus we analysed the time series photometry of 280 PNe. In total, 32 variables were found, but 5 of them turned out to be foreground objects. Another 18 objects show irregular or regular variability that is not attributable to the binarity of their central stars. Their status and the nature of their variability will be verified in the follow-up paper. Nine binary central stars of PNe with periods between 0.24 and 23.6 days were discovered. The obtained fraction for the LMC PNe is 3.3^+2.6^_-1.6_% without correcting for incompleteness. This number is significantly lower than the 12-21% derived in the analogous search in the Galactic bulge. We discuss this difference, taking into account observational biases. The lower binary fraction suggests a lower efficiency of the common envelope phase in producing close binaries in the LMC compared to the Galaxy.