I am estimating a four factor model with CFA. My 23 observed variables are likert-type and is multivariate non-normal with missing. What estimator should I use(MLR, MLF, WLSM, WLSMV....)? When I used WLSM or WLSMV, the relative fit indices were good(CFI=.966 or .90, TLI=.962 or .96) but the absolute fit index were not so good(RMSEA=.086 for both). In case of MLM or MLMV, the relative fit indices were not so good(CFI=.83 or .838, TLI=.81 for both) but the absolute fit index were good(RMSEA=.054 for both). I wonder my variables should be regarded as continuous or ordered categorical.
If your Likert variables have floor or ceiling effects which it sounds like they have, then you should use categorical data modeling using either weighted least squares or maximum likelihood estimation. The way to specify this is to put the dependent variables on the CATEGORICAL list. I suspect they should be treated as categorical based on the difference in the results by treating them as categorical versus continuous.
You should not obtain very different results with different estimators as far as parameter estimates go. Standard errors and fit statistics will be different and could result in slightly different patterns of significance. For information about the estimators implemented in Mplus, see Technical Appendices 4 and 8 and references therein and pages 482-485 of the user's guide. Also, see the general literature on maximum likelihood and weighted least squares.