A causal relationship is defined by a structural model that relates an outcome to its causes. Regression is one way of estimating the parameters of the structural causal model (there are other ways). If the structural model takes the form of a logistic regression model, then a logistic regression model is one way of recovering the true causal parameter. If the structural model takes the form of a linear regression model, then a linear regression model is one way of recovering the true causal parameter. With a binary outcome, a logistic regression model makes more sense because the the output of a logistic regression model can be interpreted as a probability, and part of the outcome-generating process may involve drawing a 0 or 1 with a given probability. With a continuous outcome, a linear regression may make more sense. The choice of model you use should depend on the form of the structural causal model you are trying to approximate. (Note that both linear and logistic regression can be used for both binary and continuous outcomes.)
What many of the posters here are arguing is that regression (logistic or linear) are not inherently causal methods; they can be used to extract a causal parameter from data, but they can be used for other purposes, too. My view is that as long as the predictor precedes the outcome temporally, regression is inherently causal, regardless of your study design. The difference is that the parameter you may estimate from your model may be a biased estimate of the causal parameter. The degree of bias depends on qualities of your design (e.g., whether you have randomized your treatment, whether you are implicitly conditioning on a consequent of treatment, whether you have collected enough variables to remove confounding). Including covariates in a linear or logistic regression is one way to attempt to remove the bias of a causal effect estimate.
When the structural causal model is perfectly reproduced by the model you specify, the causal parameter of interest will be estimated without bias. This tends not to be the case, and so estimating causal parameters using regression can leave you with a biased estimate (if bias is induced by your design). It is for this reason that other answers have been strong in claiming regression is not an inherently causal method; in a bias-inducing design, it's almost impossible to specify a regression model that correctly reproduces the structural causal model that underlies the data.
A final note: people often say correlation does not imply causation, but correlation without confounding does imply causation. Regression is one way to remove confounding. The effectiveness of logistic regression or linear regression at doing so depends on generally unknown qualities of the data-generating structural causal model.