What is Item Response Theory (IRT) called for continuous response? I would like to model my problem using something similar to Item Response Theory, but my responses are not binary. They are continuous in $[0, 1]$.
What are these models/the research field called?
 A: If you have a continuous indicator, then you would use factor analysis. Think of FA as linear regression and IRT it's logistic regression brother.
A: I agree that the accepted answer does not point towards specifically dedicated models.
See for instance:

*

*Noel, Y. & Dauvier, B. (2007). A beta item response model for
continuous bounded responses, Applied Psychological Measurement, 31,
47-73.

*Noel, Y. (2014). A beta unfolding model for continuous bounded
responses, Psychometrika, 79(4), 647-674.

*Verhelst N.D. (2019).
Exponential Family Models for Continuous Responses. In: Veldkamp B.,
Sluijter C. (eds) Theoretical and Practical Advances in
Computer-based Educational Measurement. Methodology of Educational
Measurement and Assessment. Springer, Cham.

Best,
G.
A: The accepted answer does not give models where the response is bounded between [0,1]. There are IRT models for exactly the case where the response variable is continuous, but bounded in this way. 
For example, Samejima [1] describes exactly this case. These models are sometimes called Continuous Response Models (CRM), and the case they're addressing is sometimes called Continuous Response Format (CRF). They are the standard factor analysis setup, but add a transformation to bound the responses within an interval (in the same sense that the standard IRT takes factor analysis, and adds a link so responses will be, say, 0 or 1). 
Ferrando [2] has a good paper on this, and the R library estCRM looks like it can implement it (though I haven't used it!).
[1]: Normal ogive model on the continuous response level in the multidimensional latent space
[2]: Theoretical and Empirical Comparisons between Two Models for Continuous Item Response
