You can't use delete to remove an item from an array. This is only used to remove a property from an object.

You should use splice to remove an element from an array:

deleteMsg(msg:string) {
    const index: number = this.data.indexOf(msg);
    if (index !== -1) {
        this.data.splice(index, 1);
    }        
}

Same way as you would in JavaScript.

delete myArray[key];

Note that this sets the element to undefined.

Better to use the Array.prototype.splice function:

const index = myArray.indexOf(key, 0);
if (index > -1) {
   myArray.splice(index, 1);
}

You could use Array#filter method:

food = {
  id: 1,
  name: 'Pizza',
  price: 16
};

orders = [
  { food_id: 2, table_id: 5 },
  { food_id: 2, table_id: 5 },
  { food_id: 1, table_id: 5 },
  { food_id: 3, table_id: 5 },
  { food_id: 1, table_id: 5 }
];

removeFoodOrder(food: Food): void {
  this.orders = this.orders.filter(({ food_id }) => food_id !== food.id);        
}

Edit:

Since your array allows duplicate elements and you want to remove only the first match, you could use the Array#findIndex + Array#filter methods:

const foundIndex = this.orders.findIndex(({ food_id }) => food_id === food.id);
this.orders = this.orders.filter((_, index) => index !== foundIndex);

<li *ngFor="let languague of listOfLanguagues; let i = index">

<button type="submit" (click)="removeLanguague(languague, i)" >Remove</button>

removeLanguague(languague, index){
    this.listOfLanguagues.splice(index, 1);
}