In short, the research I looked at found that propolis can have anti-plaque, anti-tartar, and antibacterial effects. Just be careful you're not allergic to it.
A pretty comprehensive review article Does Propolis Help to Maintain Oral Health? cites many different studies on all the research that's been done on the potential benefits of putting propolis in people's mouths.
In most researches propolis is used directly in the mouth in the form of ethanol- or water-based mouth rinses [27, 28, 31, 32, 34–36, 38] or in the form of a toothpaste [29, 30]. Propolis can be also used in a form of a solution to decontaminate fibres of toothbrushes .
A research carried out by Tanasiewicz et al. showed clinical effectiveness of a toothpaste and gel containing 3% ethanolic extract of propolis in a group of patients with a greater risk of gingivitis caused by dental plaque . As propolis mouth rinses and propolis-based toothpastes stop the growth of pathogens of gingivitis and periodontitis, they seem to be promising not only as preventive but also as therapeutic agents [30, 45, 48, 49]. The results of the study by Sonmez et al. showed, however, that propolis extracts in concentration that effectively reduces pathogenic organisms for periodontal diseases are cytotoxic for the gingival fibroblasts . Preventive effect of propolis on periodontal tissues includes also the slowing down of formation of precipitates of calcium phosphates and because of that, it can be used as ingredient of mouthwashes or toothpastes in order to limit the accumulation of dental plaque 
I'll cover some of the relevant referenced studies below.
: This study looks good, at least judging from the abstract. It seems like they studied humans; if 42 includes both the control and propolis group the sample size isn't great.
Propolis is bee-produced substance with pronounced anti-inflammatory effect. It is an ingredient of many drugs; it is added to toothpastes as a prophylactic component for periodontal diseases. The plaque-cleaning, plaque-inhibiting and anti-inflammatory actions of the silicate paste were studied. The study included 42 individuals in good clinical health and a minimum of 20 intact teeth - 11, 16, 24, 31, 36, 46. Control exams were performed on days 1, 7, 14, 21 and 28. The baseline values gradually decreased after each control tooth brushing with a statistically significant difference between them. The toothpaste shows very good plaque-cleaning, plaque-inhibiting and anti-inflammatory effect.
A clinical study of a silicate toothpaste with extract from propolis.
: This paper studied 80 people. Half the group (40) was assigned to test a gel and the other tested toothpaste. The toothpaste group was split in half where 20 people used propolis toothpaste and the other half used a non-propolis toothpaste of the same brand. Half of each of these groups had gingivitis, half without. So I think that that means there was 10 people in each section, which isn't great at all. People were also told what toothpaste they got which means it's not blind. The results seem to support propolis toothpaste being good against gingivitis:
Results of the research show the effectiveness of hygienic preparations with 3% content of ethanol propolis extract in both groups of patients: without pathological changes within the boundaries of the periodontium and in the case of patients endangered with the occurrence of gingivitis caused by dental plaque
Influence of Hygienic Preparations
with a 3% Content of Ethanol Extract of Brazilian
Propolis on the State of the Oral Cavity
: This study was in vitro and didn't actually test toothpastes, but it did test multiple types of propolis. Also the authors specifically mention how promising their results are for applications in toothpastes:
These results suggest that eight honeys and three types of propolis may have potential as anticalculus agents in toothpastes and mouthwashes.
Inhibition of the formation of oral calcium phosphate precipitates: the possible effects of certain honeybee products
: This study didn't test on people, but it does support propolis being good for the mouth:
The effect of propolis collected in Brazil on the adsorption of Streptococcus
sanguis to saliva-coated hydroxyapatite and on the coaggregation reaction between S.
sanguis and Fusobacterium nucleatum was studied using radio-labeled bacterial cells.
The antibacterial activity of a propolis-containing toothpaste against cariogenic and
plaque-forming bacteria was also investigated. An ethanolic solution of propolis (10 and
20 mg/mL) significantly reduced bacterial adherence to saliva-coated hydroxyapatite
(25.3% and 29.6%, p<0.05). Coaggregation reaction was also significantly inhibited
by 42.9% (10 mg/mL, p<0.01) and 56.2% (20 mg/mL, p<0.01). The toothpaste completely
inhibited the growth of 20 bacterial strains, including Streptococcus, Actinomyces and
Lactobacillus at a concentration range of 3-7 mg/mL. In addition to previously reported
effects of propolis on mutans streptococci, propolis used in the present study was
shown to possess remarkable potency to inhibit plaque formation and development.
Propolis and propolis-containing toothpaste might represent potent inhibitory agents
against dental plaque formation.
Effect of propolis and propolis-containing toothpaste on the formation of dental plaque in vitro
(Emphasis added in any quotes that have it.)