Is a glm appropriate with the dependent variable on an interval scale? Data has to meet some criteria concerning scale if you want to use a GLM. For Ordinally scaled dependent variables GLM's are clearly not appropriate for example. I couldn't find anything which speaks against an interval scaled dependent variable.
In my opinion when a dependent variable on an interval scale is used it shouldn't make a difference for an analysis if you transform the variable by adding a fixed constant to each value. I created a random data set in r to see if this is the case with GLM's. In my example I'm trying to create example data with a strictly linear response and gamma distributed errors:
set.seed(42)
# Arbitrary x-values of an imaginary covariate:
xseq <- seq(1, 10, length.out = 1000) 
# The expectance value of the dependent variable (In this case it is the
# same as the x-values because the relationship is supposed to be strictly linear):
mu <- xseq 

# Add some gamma distributed noise to mu to get the dependent variable:
shape <- 2 # Arbitrary shape parameter of a gamma distribution:
rate <- shape/mu # The rate parameter (shape divided by expectance value):
ysim <- rgamma(1000, shape = shape, rate = rate)

# A first model with an identity link function:
glm.base <- glm(ysim ~ xseq, family = Gamma(identity), start = c(0, 1))
# Predicted values for a regression line:
glm.base.fit <- fitted(glm.base)

# plot the values and the regression curve with extra space for the second model:
par(mfcol = c(1, 2))
plot(ysim ~ xseq)
lines(glm.base.fit ~ xseq, col = 'blue', lwd = 2)

# The second model with shifted values of the dependent variable:
glm.shifted <- glm(I(300 + ysim) ~ xseq, family = Gamma(identity), start = c(300, 1))
# Fitted values for a regression line:
glm.shifted.fit <- predict(glm.shifted, newdata = data.frame(xseq = xseq))

# plot the second model:
plot(ysim + 300 ~ xseq)
lines(glm.shifted.fit ~ xseq, col = 'red', lwd = 2)

# compare the coefficients of both models:
coef(glm.base)
coef(glm.shifted)

You hardly can see the difference between both lines but the coefficients reveal that it exists. For data of another structure the difference is bigger (but I'm not sure if the design of the data and the analysis is really correct ...): 
set.seed(42)
# Arbitrary x-values of an imaginary covariate:
xseq <- seq(1, 10, length.out = 1000) 
# The expectance value of the dependent variable. In this case not strictly linear:
mu <- xseq/(3+xseq)

# Add some gamma distributed noise to mu to get the dependent variable.
shape <- 2 # Arbitrary shape parameter of a gamma distribution:
rate <- shape/mu # The rate parameter (shape divided by expectance value):
ysim <- rgamma(1000, shape = shape, rate = rate)

# A first model with an inverse link function:
glm.base <- glm(ysim ~ I(1/xseq), family = Gamma(inverse), start = c(1, 0.001))
# Predicted values for a regression line:
glm.base.fit <- predict(glm.base, newdata = data.frame(xseq = xseq))

# plot the values and the regression curve with extra space for the second model:
par(mfcol = c(1, 2))
plot(ysim ~ xseq)
lines(1/glm.base.fit ~ xseq, col = 'blue', lwd = 2)

# The second model with shifted values of the dependent variable:
glm.shifted <- glm(I(300 + ysim) ~ I(1/xseq), family = Gamma(inverse), start = c(0.001, -0.00001 ))
# Fitted values for a regression line:
glm.shifted.fit <- predict(glm.shifted, newdata = data.frame(xseq = xseq))

# plot the second model:
plot(ysim + 300 ~ xseq)
glm.shifted.fit <- predict(glm.shifted, newdata = data.frame(xseq = xseq))
lines(1/glm.shifted.fit ~ xseq, col = 'red', lwd = 2)

# compare the coefficients of both models:
coef(glm.base)
coef(glm.shifted)

Judging from these examples I'd say with an interval scaled dependent variable a GLM is not appropriate. Is this correct? If yes, which test to use instead?
Thanks
 A: By saying the response is measured on an interval scale you're stipulating that the zero-point is arbitrary, so you don't then go on to model it with distributions like the gamma that have a support of $(0,\infty)$. You need to use a distribution with a location parameter to absorb additive re-scaling (as well as a scale parameter to absorb multiplicative re-scaling). 
Shunning non-linear transformations of the response implies you must also shun non-linear transformations of its conditional means. So non-linear link functions are ruled out (except, trivially, for saturated models) because a unit change in the linear predictor doesn't have an effect on the original response scale invariant to location shifts.
These two requirements leave you looking for a distribution in the exponential family with a location & scale parameter, & for a linear link function: a Gaussian GLM with identity link is the only option. But you needn't restrict yourself to GLMs when there's a clear reason not to: for the example you give a shifted gamma would be a suitable distribution to model the conditional response, even though it's not in the exponential family.
There are other approaches to regression that don't involve worrying too much about the response distribution, not least using ordinary least squares when there's no evidence of heteroskedasticity, & relying on the central limit theorem to produce reasonably good confidence intervals for parameter estimates (even if the associated prediction intervals are unreliable). Note that as an interval scale is also ordinal, the ordinal logistic regression mentioned by @Glen_b can be used for an interval-scale response, as a kind of semi-parametric model.
