The 2011 Tōhoku earthquake and tsunami was one of the biggest in modern history; the earthquake even moved Earth's axis for a very small, but measurable bit. The four years since then is not enough for hard climate data, but enough to have some insight to how the weather was influenced. So did it influence weather in any way that could be seen as "impact on climate"? If yes, how?
Local effects are definitely possible, mainly from the tsunami itself - turbulence in the air from the waves and increased evaporation from flooded areas after. However, it appears that the earthquake and tsunami has had no significant effect on the world's weather, particularly in the long term. According to the Lamont-Doherty Earth Observatory, Columbia University web-article, Can Big Earthquakes Disrupt World Weather?, a key point is that although the earthquake released a massive amount of energy, it is not enough to disrupt the energy balance of the atmosphere.
A particular point is that one of the significant effects - the axial shift from the earthquake, is, according to the article, minimal compared to natural cyclical axial fluctuations. The article also mentions that other major earthquake and tsunami events have the same minimal effects.
Although, some studies such as what gansub mentioned in comments, Impact of tsunami on Tamil Nadu Monsoon Rainfall, suggest that the tsunami itself can cause disruptions to local and regional climatic systems -particularly with rainfall patterns. This is an ongoing study by the University of Canterbury's Dr Iman Soltanzadeh research "The Atmospheric Impacts of Volcano and Large Tsunamis" where land use modifications due to a tsunami would affect local and regional climate and weather patterns.
However, the effects on the thinner upper atmosphere were a bit more pronounced according to NASA observations reported in the web-article Quake and Tsunami make waves in the Atmosphere, causing ripples in the ionosphere (~85km above the surface), as can be seen in the image below:
The quake caused acoustic and Rayleigh waves that reached the ionosphere in about 10 minutes, and the rolling motion of the tsunami caused gravity waves to propogate to the ionosphere in about 30-40 minutes, continuing westward even after the tsunami was stopped by the east coast of Japan.
The most significant aspect of NASA's findings can be interpreted that as the waves (all types) propogate through the atmosphere from the earthquake rupture and tsunami to the ionosphere, they appear to cause no significant climatic disruption to the lower near-surface and even the middle layers of the atmosphere, further emphasising the conclusions made by the Lamont-Doherty Earth Observatory that large earthquakes and associated tsunami do not cause any long term affects on weather patterns.
False! the tsunami caused major upwellings of cold subsurface waters and even slowed, temporarily stopped or shifted some ocean currents. All this led to the drought that plagued the west coast for years, until the Humbolt current resumed it's normal path. Many other recent weather anomalies since 2011 can be blamed on this Tsunami and it's lasting effects.