Do the antimicrobial properties of honey counteract the probiotic properties of yogurt?
It appears not to. Studies using 5% (w/w) clover honey had no effect on Steptococcus thermophilus, Lactobacillus delbrukeii subsp bulgaricus and probiotics Lactobacillus acidophilus and Bifidobacterium bifidum.
You are correct that honey has been found to have antimicrobial properties against a wide range of pathogens. However, these studies are done for the scenario of using undiluted honey as a wound dressing. For example, from a recent study:
For at least 2700 years, honey has been used by humans to treat a variety of ailments through topical application, but only recently have the antiseptic and antimicrobial properties of honey been discovered. ...Clinical studies have demonstrated that application of honey to severely infected cutaneous wounds rapidly clears infection from the wound and improves tissue healing.
The broad-spectrum antimicrobial properties of honey are confirmed but the exact mechanism is unclear; it could be due to the acidity, honey's osmotic effect, presence of bacteriostatic and bactericidal factors (hydrogen peroxide, antioxidants, lysozyme, polyphenols, phenolic acids, flavonoids, methylglyoxal, and bee peptides), etc.
From a review atricle,
A bacterial level greater than 105 organisms per gram of wound tissue has been found to have a deleterious effect on wound healing in surgical and chronic wounds. [This article discusses] the inherent complexities of the clinical use of medical-grade honey... and to select clinical entities in patients who may benefit from treatment with medical-grade honey, using the evidence indicators such as the Cochrane reviews.
Studies have demonstrated that the antibacterial properties of honey are more complex than just high sugar content alone (which decreases tissue fluid to which it is applied, inhibiting bacterial growth.) As referred to above, honey contains an enzyme that converts glucose to hydrogen peroxide, an antibacterial agent. Also, as you mentioned, the antibacterial properties of honey appear to vary depending on the floral source.
Honey derived from Leptospermum trees (manuka) or Echium vulgare bush (viper's bugloss) showed antibacterial properties independent of hydrogen peroxide. It is believed that another, yet undiscovered, component of honey is responsible for the antibacterial properties.
As mentioned above, phenols are significant:
Antioxidant potential was dependent of honey extract concentration and the results showed that dark honey phenolic compounds had higher activity than the obtained from clear honey.
There have been interesting studies of susceptibility of food-borne pathogen to honey, or potential for treatment of stomach ulcers with honey, but these are in vitro
Assessment of the minimum inhibitory concentration by inclusion of manuka honey in the agar showed that all seven isolates tested had visible growth over the incubation period of 72 h prevented completely by the presence of 5% (v/v) honey.
Even in vitro, significant minimum inhibitory concentration started at about 9%, and against some organisms, 25%.
One paper isolating methylglyoxal (MGO) (and other agents) found Manuka (Leptospermum scoparium) honey had very high amounts of MGO, up to 100-fold higher compared to conventional honeys.
Whereas most of the honey samples investigated showed no inhibition in dilutions of 80% (v/v with water) or below, the samples of Manuka honey exhibited antibacterial activity when diluted to 15–30%, which corresponded to MGO concentrations of 1.1–1.8 mM. This clearly demonstrates that the pronounced antibacterial activity of New Zealand Manuka honey directly originates from MGO.
All this is to say concentration (as well as the source of the honey) matters. If one of the most consistently antibacterial honeys (manuka)has to be present in concentrations of at least 15% to be significantly antibacterial in vitro, unless you're chugging manuka honey, I don't think it will have a deleterious effect on your intestinal flora.
Growth and Acid Production by Lactic Acid Bacteria and Bifidobacteria Grown in Skim Milk Containing Honey
Antimicrobial Properties of Honey
Use of Honey in Wound Care: An Update
Antioxidant and antimicrobial effects of phenolic compounds extracts of Northeast Portugal honey
Susceptibility of Helicobacter Pylori to the Antibacterial Activity of Manuka Honey
The antibacterial properties of Malaysian tualang honey against wound and enteric microorganisms in comparison to manuka honey
Identification and quantification of methylglyoxal as the dominant antibacterial constituent of Manuka (Leptospermum scoparium) honeys from New Zealand