I understand that a monopolist optimally sets a two-part tariff by pricing at marginal cost and then extracting CS as a lump sum fee. However, wouldn't the lump sum fee induce a decrease in consumer demand? Is the typical assumption made in two-part tariff problems that there is no income effect?
In the typical textbook treatment of the two-part tariff model, income effect is ignored/assumed to be negligible.
Goldman, Leland, and Sibley (1984) provided detailed comparison of optimal nonlinear pricing strategies (of which two-part tariff is a special case) with and without income effect. Their conclusion is
When [income effects] are absent, the aim of optimal pricing is to raise a given amount of profit with the least possible distortion to individual's consumption plans. Therefore increments of consumption are priced according to the Ramsey rule. When income effects are present, the optimal pricing problem is equivalent to the optimal taxation problem, and the goal is optimal income redistribution.
In typical textbook two-part tariff there won't be an income effect because either demand will be just assumed to be given by some function or even if there is a utility function it won't feature budget constraint (e.g. see examples of such simple problems in Belleflamme and Peitz, Industrial Organization: Markets and Strategies pp 227-234).
In such cases the tariff won't have any effect on demand itself because it is a sunk cost from the perspective of consumer. Such assumptions are appropriate for items that are not too expensive (e.g. razors, club entry etc.) but would not be appropriate for more expensive items.
However, once we add budget constraint then there will be income effects and demand will be affected by the fee. For an example of such model you can see Ng & Weisser (1974). This changes the problem somewhat but authors show that even in such cases two-part tariff is still often optimal.