var arr = [
[1, 2, 3, 4, 5, 6, 7],
[4, 3, 2, 55],
["", "", "", ""]
];
const newArr = arr.filter(n => n.some(m => m !== ''));
arr.reduceRight((_, n, i, a) => n.every(m => m === '') && a.splice(i, 1), null);
// или
arr.splice(0, arr.length, ...arr.filter(n => n.some(m => m !== '')));
// или
let numDeleted = 0;
for (let i = 0; i < arr.length; i++) {
arr[i - numDeleted] = arr[i];
numDeleted += arr[i].every(n => n === '');
}
arr.length -= numDeleted;