From a sales perspective, posting rates can dequalify prospects without them having to reach out to you. With that said, posting rates helps prospects figure out potential costs before they decide to contact you. Whether you should post rates in the app really depends on whether you're receiving sufficient leads as is and whether you could increase conversion by including the rate in the app.
What's your rationale of not putting rates in the app? I haven't seen our app but do you think it really portrays the unique value-add of your paid consulting service?
Also, you mentioned that the app gets a few downloads here and there. It sounds as though there's not many new people downloading the app. If you don't have a lot of new users downloading the app, assuming conversion rates stay the same (i.e. for every 20 downloads, 1 email contact), the time it takes to receive an email will increase.
Taking a step back, it sounds as though you're using the app, in part, as a marketing channel for new leads. Perhaps it's not the best medium? You may be able to use the app to direct users to a website that's more feature rich. It's easier to showcase your past successes and capabilities on a website than it is on a mobile app. You can also reach a greater audience and perhaps get more leads.
If anything, I don't think there's a hard and fast rule of whether it makes sense to include prices. Some consultants will include their prices while others don't. Many people assume that when prices aren't included, the actual per-hour cost will be pretty high (i.e. "if you have to ask, you can't afford it" mentality). However, if you have establish enough brand value, then you'll be successful even without including the price.
At the same time, including the price isn't necessarily tacky. Including prices helps to weed out prospects but also makes it difficult to change prices (especially up).