The speed of light is a cosmic speed limit of sorts, so if the universe is expanding such that neither celestial object is moving faster than the speed of light, but the DISTANCE between the two is growing faster than the speed of light, then we have a situation where basically the star will be invisible because the photons simply can't cover the distance while abiding by the cosmic speed limit (speed of light).
So the short answer is no, there are cases where we would never be able to detect stars. Their photons will simply be lost traveling in space, never reaching a destination (This is why faster than light travel is necessary if we ever hope to even reach the outskirts of our own galaxy, let alone others).
To go one step further, we know for a fact that as galaxies continue to expand away from one another, we will eventually reach a point where the galaxies we can now easily see, will slowly start dimming and eventually disappearing from the night sky, until the only stars we can see are ones specifically within our galaxy. We also can't know if there are galaxies that this has already occurred with, which is why people generally talk about the universe in terms of our observable universe, since there could be plenty of things out there that we simply cannot see.
The Alcubierre Drive is a speculative faster-than-light engine that relies on expanding and contracting space-time to achieve a faster-than-light traveling speed between objects, even though the space craft would never actually travel faster than light. Similar to how the earth and star could move (relatively speaking) faster than the speed of light away from each other, except this work to close the distance faster than the speed of light.
Another quick example is if a train is moving at nearly the speed of light, and someone starts running forward in the train, the train's on-board time will slow down to ensure the cosmic speed limit is enforced and that the person cannot move faster than the speed of light.
Physics blows my mind.
And if I made any mistakes, please correct me! Thanks!