Seiji and Mika share an odd relationship. Seiji, at first, had no interest in Mika at all, saving her and Anri from a few thugs on a whim. This act sparked Mika's interest, however, and she began to stalk him over the course of several weeks. After Namie has Mika's face altered to look like Celty's, he acts much more affectionate toward her, but this is because he thinks she is Celty's head come to life and not Mika herself. Once it is revealed that Mika is not Celty's head, Seiji is shocked but decides to accept her love, at least until the real head can be recovered. From then on, to the casual observer, the two of them maintain a fairly normal (if overly affectionate) romance throughout the series.
While his reasons are rather twisted, Seiji does treat Mika as a genuine girlfriend, although he does seem to have a rather passive attitude most of the time. He has been shown to physically stand up for Mika at various points, such as in the first season OVA, when he punches a thug in the face for "attempting to thwart their love" (really, the thug just bumped into him). In the light novels, he appears to be steadily showing signs of accepting Mika for who she is, as opposed to simply using her as a replacement for Celty's head.
Mika, in contrast, is completely obsessed with Seiji, going to extreme lengths to please him and just be around him. She goes so far as to radically change her appearance and fill his room to the brim with bugs and cameras. Seiji takes Mika's overbearing personality with a calm demeanor and has grown used to it over time. Mika frequently gushes about how in love they are, and Seiji rarely (if ever) objects to these statements. In the light novels, Mika is shown to genuinely care about Seiji's well-being and is willing to go to extreme lengths to keep him out of harm's way, using every method she can to keep him distanced from chaotic events within Ikebukuro, such as the Toramaru incidents or the sudden unrest in the Dollars. She does this because she knows that Seiji still feels somewhat guilty about attacking Mikado during the first Dollars meeting and, given his generally chivalrous nature, knows he would try to help Mikado and put himself in danger in the process.
The conflicts between the two are few but serious. While Seiji does act affectionate to a certain extent, it is clear he is only using Mika as a replacement for Celty's head, which he is still actively searching for. Likewise, Mika (in the manga and anime) expresses a desire to somehow consume Celty's head to make it a part of herself and force Seiji to stay with her. These aren't so much conflicts in their relationship so much as they are twisted aspects of their relationship; however, they will likely lead to some tension as the series continues.
On the other hand, the main conflict would be the third party - Seiji's sister, Namie. Technically, there is a love triangle in Seiji and Mika's relationship, with Namie being Mika's competitor. Though Seiji is fully aware of his older sister's warped sexual attraction towards, he does not seem to be overall under any stress or conflicted. Instead, he merely ignores this, and rather attempts to maintain a normal sibling relationship with Namie.