Do jet contrails impact weather?
Short answer

Based on past observations and a study led by Patrick Minnis from NASA, contrails could result in increasing the temperature over the ground by 1°C every 30 years over the US.

From a study by German researchers Ulrike Burkhardt and Bernd Kärcher, the effect of contrails is comparable in other areas with a similar level of air traffic.

Because the air traffic is said to be about to increase very quickly, the effects of contrails on the temperature may increase too.



Contrails and natural cirrus: So similar that studies are difficult

Contrails are like cirrus, and actually some just become persistent cirrus. Sometime it's difficult to tell which one is a condensation trail, which one is a natural cloud. This makes more difficult the study of contrails effects, and introduces many uncertainties that need to be reduced:


(source)

The current studies take into account only the young contrail-cirrus, because they are still relatively easy to differentiate from natural cirrus, due to their linear shape. Most aged contrail-cirrus are not differentiable and are not counted as aircraft generated. This means the current results potentially under-estimate the effects of contrails.

Pertaining to the discussion, contrail-cirrus as clouds:


Induce a cloud-feedback, i.e. the coupling between cloudiness and surface air temperature.
Induce a radiative forcing, they increase the portion of the energy not re-emitted to space. The quantity is expressed in mW/m².


Increasing the air temperature has a link with increasing ground or oceans temperature, but the two notions mustn't be mixed. However cirrus are suspected to play a role in the global warming, because they could introduce a positive feedback into this warming.



Study: Coverage and effects of contrail-cirrus

Global radiative forcing from contrail cirrus published in Nature in 2011 was a study to evaluate the contrail-cirrus coverage, and its impact. Conclusion:


  Aviation makes a significant contribution to anthropogenic climate
  forcing. The impacts arise from [...] and from changes in cloudiness in the
  upper troposphere. An important but poorly understood component of
  this forcing is caused by ‘contrail cirrus’—a type of cloud that
  consist of young line-shaped contrails and the older irregularly
  shaped contrails that arise from them.


The estimated effect is depicted on this map (mW/m²) from the study:  

  

Net effect of contrails (CIC) warming compared to other aviation factors of warming:  


(source)  



Study: Effect of contrail-cirrus on US climate

NASA conducted a study in 2004: Patrick Minnis, J. Kirk Ayers, Rabindra Palikonda, and Dung Phan -- Contrails, Cirrus Trends, and Climate.

This study was analyzed in Science Daily: Clouds caused by aircraft exhaust may warm the US climate:


  Using published results from NASA's Goddard Institute for Space
  Studies (New York) general circulation model, Minnis and his
  colleagues estimated contrails and their resulting cirrus clouds would
  increase surface and lower atmospheric temperatures by 0.36 to 0.54 degrees Fahrenheit per decade.




Aviation radiative forcing in 2000: An update on IPCC (1999)

This study was included in a new one by R. Sausen et al. in 2005. It shown the radiative forcing by aircraft could have been slightly underestimated:


Source: Aviation radiative forcing in 2000: An update on IPCC (1999) - R. Sausen



The US surface temperature is increasing by 1°C every 30 years: I would say this is very significant for plants, life, water cycle, oceanic currents, etc.

Also look at @Simon good answer about the increase of the difference between night and day temperatures, by more than 1°C, during the 3 days civil air traffic was forbidden on September 2001. This increase seems to confirm contrail-cirrus limit cooling on night time.