The answer is probably that it depends on the specific model of the scale. If you care strongly about precision, you should obtain a set of reference weights and conduct a series of experiments to establish that your protocol of poweron-container-tare-weigh satisfies your requirements for precision, accuracy, and linearity in the region that you're interested.
All devices with a digital readout have a minimum uncertainty of $\pm1$ unit in the final displayed digit. That's because the analog-to-digital converter that talks to the display must have some hysteresis (usually some version of a Schmitt trigger) so that electronic noise doesn't sometimes make the last digit flicker between adjacent values. That hysteresis means there is a "dead zone" within each displayed value where an adjacent value would be a better approximation of the signal, but the design engineers have decided not to trust that value tell the display to change. It's impossible to know the width of this dead zone without getting into the nitty-gritty of the electronics.
Of course a device can have a worse uncertainty than one unit in the least significant digit. I have a bathroom scale that displays my weight to the nearest tenth of a pound — but it only displays even tenths digits (0.2, 0.4, but never 0.3). Furthermore, that particular scale has terrible reproducibility: if I get off and get back on, I may see the same value or I may see a value that's two or three pounds away. (That problem hit me this summer when I was trying to determine whether a geriatric cat was losing weight. Twenty years ago with an analog scale, using myself as a tare was a robust way to weigh a small animal. But while the two-pound uncertainty in my digital scale is a 1% uncertainty for me, it was a 40% uncertainty for the cat — a problem of catastrophic cancellation.)
In a high-quality scale where digitization noise is the primary source of uncertainty, it's possible that an internal tare could improve your precision by storing the tare value (whether digital or analog) with a higher precision than is displayed. In that case the discretization noise would contribute to your signal once, when you weigh your object-plus-container, rather than contributing separately when you weigh your object and when you weigh your container. But as with so many error-analysis problems, it can be a lot of work to convince yourself that you've correctly identified what you don't know.