I could find two answers: no and yes. The no was marketed by manufacturers of large air purifiers containing full-blown HEPA while the yes by the below research on asthma, disabilities and seasonal affective disorder (SAD). Because people allergic to birch pollen are more likely to develop such disabilities unless proper interventions, this material is relevant to people having symptoms due to birch pollen. If birch pollen creates symptoms such as shortness of breath, itching, tiredness, coughing or other asthmatic symptoms, the benefits of negative ionizer can surpass the cons such as ozone particles generated. If you have bad asthma, the ionizer may worsen the symptoms or you just need to select a negative ionizer certified not to generate too much ozone.
As for ventilation, the optimal set-up seems to be to have two systems: a HEPA/cleaner/cooling/filtering system and then small negative ionisers around in the working places. The first is here while the second is discussed here. As for now, the optimal system seems to contain the negative ioniser besides the full-blown HEPA system with filters. There may be products on the market containing both, not sure about their effectiveness. As a summary, yes -- negative ioniser is effective against birch pollen -- if the negative ioniser combined with proper air-ventilation setup.
P.s. Of course, the ioniser does not replace waterfalls and Aloe Vera Barbadensis Miller, Stockton plantations that can alleviate allergic symptoms.
The ionizer can create ozone which can worsen asthamatic symptoms due to pollen. Also, it does not attract dust, pollen and allerges by the video.
As collected in a more general earlier thread, ionizers generating ozone are in Air Cleaners to Avoid. This video was on marketing a purifier with full-blown HEPA and other properties -- I would not trust it totally.
Negative ionizers on Asthmatic symptoms
This is a picturesque writing where negative ions are compared to waterfalls and observation that many asthmatic people can alleviate their symptoms near oceans and waterfalls where negative ions are generated. Negative ions are smaller than positive ions. When pollen particles that are relatively large move they create abundance of positive charge in the environment. The principle is to attract the positive particles by the negative ioniser like waterfalls that absorbs the pollen. The writing concludes
Sufferers of allergies and asthma who have experienced these positive effects can attest to the fact that negative ions provide exceptional relief. In fact, anyone who has been exposed to a negative ion generator can benefit from these charged particles, just as if breathing in the ocean air.
where the tune is clearly positive and encouraging for the negative ionizers.
Negative ionizers on Seasonal affective disorder (SAD)
Because allergic people are more likely to become depressed, the following findings on seasonal affective disorder with negative ionizer can be interesting
"Treatment with a high-density negative ionizer appears to act as a specific antidepressant for patients with seasonal affective disorder. The method may be useful as an alternative or supplement to light therapy and medications." (1995)
"Negative ion therapy: Negative ions have been shown to have a positive effect on SAD. You can purchase ion generators, or even find light boxes that have built in ion generators. Most people have the ion generator near their bed so they could treat themselves with this while they sleep."
where the sources did not find side-effects due to the small amount of ozone generated by the devices.
Are air purifiers and ionizers effective? (more general)
What methods exist for filtering air to alleviate allergies? (more general)