We don't really know.
Climate models agree that the feedback is profound. Significant. Unfortunately, they do not agree about the magnitude of the feedback. Nor about the sign. The problem is that there are competing feedbacks. Clouds act similar to greenhouse gases, because they absorb and re-emit terrestrial radiation. But they also reflect solar radiation. These feedbacks compete, and both have a large uncertainty. The net effect is the difference between the two, and the difference between two quantities with a large uncertainty, has a very large uncertainty.
This is a major topic of research and it has been a major source of uncertainty in global climate models for... decades. A major paper on the topic, a bit old but still mostly true, is:
Quoting from this article, published in 2005:
Thus we have no clear theory that suggests the accumulated effects of cloud feedbacks are in any way a function of global-mean temperature or, as posed, ΔT.
And from IPCC AR4 WG1, published in 2007:
Cloud feedbacks remain the largest source of uncertainty in climate sensitivity estimates.
The most state-of-the-art knowledge at the time of writing is summarised in IPCC AR5 WG1 chapter 7 (PDF), which stated in 2013:
Owing to the large magnitudes of the SWCRE [short wave cloud radiative effect] and LWCRE [long wave cloud radiative effect], clouds have the potential to cause significant climate feedback (Section 7.2.5). The sign of this feedback on climate change cannot be determined from the sign of CRE [cloud radiative effect] in the current climate, but depends instead on how climate-sensitive the properties are that govern the LWCRE and SWCRE.
Does that mean we can say nothing? Well, it's not quite so pessimistic. There are many attempts to answer this question. The figure below shows estimates for the regional distribution of the shortwave and longwave cloud radiative effect, again from AR5 WG1:
Figure 7.7 | Distribution of annual-mean top of the atmosphere (a) shortwave, (b) longwave, (c) net cloud radiative effects averaged over the period 2001–2011 from the Clouds and the Earth’s Radiant Energy System (CERES) Energy Balanced and Filled (EBAF) Ed2.6r data set (Loeb et al., 2009) and (d) precipitation rate (1981–2000 average from the GPCP version 2.2 data set; Adler et al., 2003). Figure reference: Boucher, O., D. Randall, P. Artaxo, C. Bretherton, G. Feingold, P. Forster, V.-M. Kerminen, Y. Kondo, H. Liao, U. Lohmann, P. Rasch, S.K. Satheesh, S. Sherwood, B. Stevens and X.Y. Zhang, 2013: Clouds and Aerosols. In: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change [Stocker, T.F., D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P.M. Midgley (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.
This suggests the effect is negative, but the debate goes on. To be continued!