```python
def tile_renders(requests: list[tuple[int, str]], fresh_for: int, size: int) -> list[tuple[int, str]]:
    if not isinstance(fresh_for, int) or not isinstance(size, int):
        raise ValueError("fresh_for and size must be positive integers.")
    
    if fresh_for <= 0 or size <= 0:
        raise ValueError("fresh_for and size must be positive integers.")
    
    cache = sorted([None] * size)
    renders = []
    
    for tick, name in requests:
        if tick < 0:
            raise ValueError("tick cannot be a negative number.")
        
        if tick not in range(size):
            # Drop the oldest render that will not satisfy this request
            while cache and tick - fresh_for >= len(cache) - 1:
                oldest_render = min(cache)
                renders.append((oldest_render, name))
                cache.remove(oldest_render)
        
        # Store the current render in the cache
        if tick - fresh_for < len(cache):
            cache[tick - fresh_for] = name
        else:
            raise ValueError("Cache is full.")
    
    return renders
```