SuNseTgReeN

Во-первых, стоит сделать поле модели project скрытым полем формы:

class AddResponseForm(forms.ModelForm):
    class Meta:
        model = Response
        fields = ('body', 'project')
        widgets = {'project': forms.HiddenInput()}

А потом в views.py:

form = AddResponseForm(request.POST)
if form.is_valid():
    response = form.save(commit=False)
    response.user = request.user
    response.save()