No, because citation is a matter of transparency to scholarship generally...
No, one does not "thank" someone for citing one's paper. An academic author is obliged to cite a paper when he/she is either placing reliance on or referring to its ideas, content, or argument. An academic author does not cite a paper simply because he/she wants to promote its author. To "thank" such an author would be an insult, in that it undermines the principle of separating the work from the person.
Decisions on whether to cite something should be independent of personal sentiment. Authors often cite work with which they disagree, sometimes profoundly. Sometimes, an author will find himself/herself having to cite material with he/she finds repulsive, especially in disciplines such as history or sociology (e.g.: a scholar writing about Nazi Germany and/or the history of racism may have to cite Mein Kampf -- do you think this scholar would want to be "thanked" by neo-Nazis or eugenicists? On the contrary, if such a scholar were "thanked" by such groups, he/she would probably be worried that he/she had somehow endorsed racist ideology).
...but you can still contact the author if you have something to say
One can, nonetheless, still make contact with authors who cite one's work. The purpose of such contact, however, should be to discuss the ideas, and not to imply any sense of personal debt/obligation (citations are an obligation to the academic community at large, not to the individual being cited). Appropriate examples include (but are not limited to):
- clarifying any misunderstandings;
- responding to criticism or counterarguments;
- asking for further details of each other's methodology/research/data;
- expressing an interest in or willingness to collaborate; and
- any other reasonable request/proposal that furthers scholarship.