To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the given information.

Let's define:
- $x_1$ as the number of dining tables produced,
- $x_2$ as the number of coffee tables produced.

The profit on each dining table is $200, and on each coffee table is $100. Therefore, the objective function to maximize profit ($P$) can be represented as:
\[ P = 200x_1 + 100x_2 \]

Given constraints are:
1. The total monthly demand for both types of tables combined will not exceed 200 units: 
\[ x_1 + x_2 \leq 200 \]
2. The monthly manufacturing budget on tables is $20,000, with dining tables costing $250 to make and coffee tables costing $150:
\[ 250x_1 + 150x_2 \leq 20000 \]
3. Non-negativity constraints since the number of tables cannot be negative:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'number of dining tables'), ('x2', 'number of coffee tables')], 
    'objective_function': '200*x1 + 100*x2', 
    'constraints': ['x1 + x2 <= 200', '250*x1 + 150*x2 <= 20000', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we'll write the following code:

```python
from gurobipy import *

# Create a new model
m = Model("Furniture_Problem")

# Define variables
x1 = m.addVar(name='x1', lb=0)  # Number of dining tables
x2 = m.addVar(name='x2', lb=0)  # Number of coffee tables

# Set the objective function: Maximize profit
m.setObjective(200*x1 + 100*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + x2 <= 200, name='total_tables')
m.addConstr(250*x1 + 150*x2 <= 20000, name='budget')

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of dining tables: {x1.x}")
    print(f"Number of coffee tables: {x2.x}")
    print(f"Maximum profit: ${200*x1.x + 100*x2.x:.2f}")
else:
    print("No optimal solution found.")
```