I'm not sure why you have this hesitance against using accent colors in your app, especially in an app which I assume will display data records of different categories (expenses/income) in a timeline. The use of color coding to visually differentiate between the records sounds like a slam dunk for me.
Don't get me wrong, I'm not encouraging you to use a blue-ish/red-ish background with dark green text on top of it, not at all. Stick with a light white background with dark text for content, but please introduce some accent colors for CTA's/charts(which I think a financial app should have)/category titles/buttons/etc... Both for making interactive elements pop but also to make it visually appealing.
You say you want to use white and different shades of gray for your palette and then use red and yellow for errors and warnings. But unless you expect the user to trigger these errors and warnings regularly (which will probably make them abandon the app anyway) I think the app will be a visually bland experience. It can still be useful in the sense that it tells the user what they need to know, but a flat uninviting UI will not entice the user to explore the app whatsoever and any features you introduce in the future may therefore go unnoticed because the app tells the user in its visual representation "I'm uninteresting, I probably have nothing more to give you. Please, stop looking at me...".
To make the app successful you need to capture the aesthetic elements for your app.